# Ansätze for Scattering Amplitudes from $p\kern 0.80002pt\text{-adic}$
Numbers and Algebraic Geometry

Giuseppe De Laurentis<EMAIL_ADDRESS>b Ben Page
<EMAIL_ADDRESS>Physikalisches Institut, Albert-Ludwigs-Universität at
Freiburg, Hermann-Herder.Str. 3, D-79104 Freiburg, GermanyTheoretical Physics
Department, CERN, Geneva, Switzerland

###### Abstract

Rational coefficients of special functions in scattering amplitudes are known
to simplify on singular surfaces, often diverging less strongly than the naïve
expectation. To systematically study these surfaces and rational functions on
them, we employ tools from algebraic geometry. We show how the divergences of
a rational function constrain its numerator to belong to symbolic powers of
ideals associated to the singular surfaces. To study the divergences of the
coefficients, we make use of $p\kern 0.80002pt\text{-adic}$ numbers, closely
related to finite fields. These allow us to perform numerical evaluations
close to the singular surfaces in a stable manner and thereby characterize the
divergences of the coefficients. We then use this information to construct
low-dimensional Ansätze for the rational coefficients. As a proof-of-concept
application of our algorithm, we reconstruct the two-loop $0\rightarrow
q\bar{q}\gamma\gamma\gamma$ pentagon-function coefficients with fewer than
1000 numerical evaluations.

††preprint:  FR-PHENO-2022-03, CERN-TH-2022-014

##  1 Introduction

Precise theoretical predictions for collider experiments rely on increasingly
higher order and higher multiplicity calculations of scattering amplitudes.
The standard method of computation is to express the amplitudes in a basis of
dimensionally-regulated master integrals, reducing the calculation to
determining the associated prefactors. These are rational functions of the
external kinematics and of the dimensional regularization parameter. However,
due to the algebraic complexity of both intermediate stages and final results
of analytic calculations, this poses a considerable challenge. To combat the
difficulty of rational prefactor computations, in recent years it has become
commonplace to compute loop amplitudes numerically over so-called “finite
fields” vonManteuffel:2014ixa ; Peraro:2016wsq , and subsequently obtain the
analytic form of the result by making use of an appropriate Ansatz. By now
there exist a number of advanced approaches for fitting specialized Ansätze
from numerical evaluations. In cases where the target function is a rational
function of an independent set of variables, then there exist “functional
reconstruction” algorithms for an arbitrary number of variables
vonManteuffel:2014ixa ; Peraro:2016wsq ; Klappert:2019emp ; Klappert:2020aqs .
In the multivariate case, these approaches reduce rational function
interpolation to the simpler polynomial case where either Newton
Peraro:2016wsq  or Vandermonde Klappert:2020aqs ; Abreu:2021asb  approaches
are used. By now these are sufficiently well understood that there exist
public implementations Peraro:2019svx ; Klappert:2019emp ; Klappert:2020aqs .
An important recent success of the Ansatz approach is the calculation of a
plethora of two-loop, five-point scattering amplitudes – both for fully
massless configurations Abreu:2018aqd ; Chicherin:2018yne ; Abreu:2019rpt ;
Chicherin:2019xeg ; Badger:2018enw ; Abreu:2018zmy ; Abreu:2019odu ;
Abreu:2021oya ; Badger:2019djh ; Chawdhry:2019bji ; Abreu:2020cwb ;
Chawdhry:2020for ; Agarwal:2021grm ; Agarwal:2021vdh ; Chawdhry:2021mkw ;
Badger:2021imn  and for configurations involving one massive particle
Badger:2021nhg ; Abreu:2021asb ; Badger:2021ega ; Badger:2022ncb .
Furthermore, we have also seen ground-breaking computations of three-loop
four-point amplitudes made possible by these tools Caola:2020dfu ;
Caola:2021rqz ; Bargiela:2021wuy .

In this work, we focus on an important problem found when applying the Ansatz
formalism to processes with a large number of scales. Specifically, as one
considers scattering amplitudes that depend on an increasing number of scales,
the complexity of functional reconstruction approaches grows exponentially.
For example, the Ansätze used in the computation of the two-loop finite
remainders for five-parton scattering required $\mathcal{O}(10^{5})$
evaluations Abreu:2018zmy , while those for four partons and a
$W\text{-boson}$ required $\mathcal{O}(10^{6})$ evaluations Abreu:2021asb .
Despite these examples, there is growing evidence that more compact Ansätze
for rational prefactors should exist. Firstly, if we look towards highly
supersymmetric theories, we see that an Ansatz consisting of leading
singularities made it possible to construct the full-color, two-loop, five-
point amplitudes in $\mathcal{N}=4$ super-Yang-Mills theory with only 6
numerical evaluations Abreu:2018aqd ; Chicherin:2018yne . Secondly, in Ref.
Abreu:2018zmy , it was observed that, when amplitudes are expressed in a basis
of “pentagon functions” Gehrmann:2018yef , the denominators of the rational
prefactors can be derived from the symbol alphabet of the associated
integrals. This observation has led to efficient algorithms for determination
of the denominator factors Abreu:2018zmy ; DeLaurentis:2019phz ;
Heller:2021qkz . Finally, it has been observed that two-loop, five-point
amplitudes in quantum chromodynamics (QCD) simplify when the rational
prefactors are cast in different types of partial-fraction decompositions, see
Refs. Abreu:2019odu ; DeLaurentis:2020qle ; Badger:2021nhg . The Leĭnartas
representation leinartas1978factorization ; raichev2012leinartas ; Pak:2011xt
; Meyer:2016slj  has received particular attention with a number of algorithms
for its computation Abreu:2019odu ; Boehm:2020ijp ; Heller:2021qkz .

Given this large body of evidence for the existence of compact Ansätze for
rational prefactors, our aim is to develop an approach to algorithmically
construct such Ansätze and thereby enable the analytic computation of two-
loop, multi-scale amplitudes with dramatically fewer numerical evaluations
over finite fields. To this end, we wish to exploit the well-known fact that
gauge-theory amplitudes admit more compact representations when expressed in
terms of spinor-helicity variables. This leads us to develop a framework based
on the approach of Ref. DeLaurentis:2019phz , which has been applied to a
number of other amplitudes DeLaurentis:2019vkf ; Budge:2020oyl ;
Campbell:2021mlr . In this approach, one takes a perspective on the
organization of rational prefactors based on their behavior on singular
surfaces. Specifically, rational prefactors are studied numerically near
surfaces where one or more of the denominator factors vanish and this
information is incorporated into an Ansatz. In order to set up this approach
algorithmically, we formalize a number of its ingredients using methods from
computational algebraic geometry. These methods have already found ample
application in the scattering-amplitude literature (see, for example,
applications to integrand reduction Badger:2012dp ; Zhang:2012ce ;
Mastrolia:2012an  and integration-by-parts relations Gluza:2010ws ). Firstly,
we interpret the spinor-helicity formalism in the language of algebraic
geometry, allowing us to use the tools of Gröbner bases to understand and
solve the problem of constructing linearly-independent polynomials of spinor
brackets. Secondly, we discuss how singular surfaces often have multiple
branches and we show how to systematically identify these branches by
constructing the primary decomposition of an ideal associated to the singular
surface. Thirdly, we show how the behavior of a rational function when
approaching surfaces where multiple denominators are singular is encoded in
its analytic structure. Specifically, we show that the numerator of the
rational function must belong to a certain ideal, controlled by the geometry
of the singular surface. The relevant tool is provided to us by the
Zariski–Nagata theorem Zariski1949 ; Nagata1962 ; EISENBUD1979157 , which
tells us to consider the so-called “symbolic power” of an associated ideal.

In order to determine the singular behavior of the rational prefactors, we
introduce a new numerical tool. Modern methods for two-loop amplitude
calculation rely on the absence of precision loss when working over finite
fields. However, finite fields lack a concept of scale separation that is
required to probe singular configurations. To this end, we work in a middle
ground provided to us by number theoretical techniques: the $p\kern
0.80002pt\text{-adic}$ numbers (see Ref. Gouvea1997  for an introduction).
While these objects are a rich source of number theory, with their own notion
of calculus, here we will only scratch the surface and use their properties as
a field. One can regard them as bridging the gap between finite fields and
floating-point numbers: $p\kern 0.80002pt\text{-adic}$ numbers have natural
expansions in powers of a prime $p$, and the first digit in such an expansion
behaves like a finite field. This set of numbers comes associated with a
concept of size, which allows us to perform numerical studies in singular
configurations. At the same time, by working with large primes $p$, there is a
low probability of numerical $p\kern 0.80002pt\text{-adic}$ calculations
involving a spurious scale hierarchy in intermediate stages. This makes it
possible to control accidental precision loss in numerical computation.
Combining this numerical tool with the algebro-geometric understanding, we
then present an algorithm for the construction of Ansätze for rational
prefactors, that can in principle be automated and applied to the computation
of novel two-loop scattering amplitudes.

This article is organized as follows. In Section 2, we present algebro-
geometric tools that allow us to understand functions on spinor space and
perform a systematic study of the singular varieties of rational prefactors.
Next, in Section 3, we give an introduction to $p\kern 0.80002pt\text{-adic}$
numbers and explain how to generate numerical phase-space points on or near
singular varieties. Thereafter, in Section 4 we collect the theoretical work
of the previous section into an algorithm to generate compact Ansätze which
leverage the singularity information. In Section 5 we make an example
application of the algorithm to the $0\rightarrow q\bar{q}\gamma\gamma\gamma$
finite remainder coefficients at two loops. Finally, we summarize and conclude
in Section 6.

##  2 Algebraic Geometry and Spinor Space

Scattering amplitudes are transcendental functions of the external kinematics,
which are typically evaluated on the set of four-momenta associated to the
external states. For an $n\text{-point}$ process, these are a collection of
momenta $\\{k_{1},\ldots,k_{n}\\}$ which satisfy on-shell and momentum-
conservation relations, that is

$k_{i}^{2}=m_{i}^{2}\quad\;\mathrm{and}\quad\;0=\sum_{i=1}^{n}k_{i}^{\mu}\,,$
(1)

where we use the all-outgoing convention. In the case of massless scattering,
where $m_{i}=0$, it is natural to employ spinor variables instead of
Mandelstam variables to describe scattering amplitudes. The connection between
a massless four momentum $k_{i}$ and a pair of Weyl spinors
$(\lambda_{i},\tilde{\lambda}_{i})$, is made through the relation

$k_{i\mu}\sigma^{\mu\dot{\alpha}\alpha}=\tilde{\lambda}^{\dot{\alpha}}_{i}\lambda_{i}^{\alpha}\,,$
(2)

where $\sigma^{\mu\dot{\alpha}\alpha}=(\mathds{1},\vec{\sigma})$ denotes an
Infeld–Van der Waerden symbol, and $\vec{\sigma}$ are the three Pauli
matrices. We take the metric on spinor space to be the $2\times 2$ Levi-Civita
symbol
$\smash[b]{\epsilon^{\alpha\beta}=\epsilon^{\dot{\alpha}\dot{\beta}}=i\sigma_{2}}$.
The metric with lowered indices is then
$\smash[b]{\epsilon_{\alpha\beta}=\epsilon_{\dot{\alpha}\dot{\beta}}=(i\sigma_{2})^{T}}$,
such that
$\smash[b]{\epsilon_{\alpha\beta}\epsilon^{\beta\gamma}=\delta_{\alpha}^{\gamma}}$,
where $\delta$ is the Kronecker delta. Raising and lowering of the indices is
achieved by contraction with the metric, that is
$\smash[b]{\lambda_{i\alpha}=\epsilon_{\alpha\beta}\lambda_{i}^{\beta}}$ and
$\smash[b]{\tilde{\lambda}_{i\dot{\alpha}}=\epsilon_{\dot{\alpha}\dot{\beta}}\tilde{\lambda}_{i}^{\dot{\beta}}}$.
Spinor-helicity variables trivialize on-shell relations, while momentum
conservation becomes a quadratic relation

$0=\sum_{i=1}^{n}\tilde{\lambda}_{i}^{\dot{\alpha}}\lambda_{i}^{\alpha}\,.$
(3)

Invariant quantities can be built by contracting the spinors in so-called
spinor brackets. We define them through the following contractions

$\displaystyle\langle
ij\rangle=\lambda^{\alpha}_{i}\lambda_{j\alpha}\quad\;\text{and}\quad\;[ij]=\tilde{\lambda}_{i\dot{\alpha}}\tilde{\lambda}^{\dot{\alpha}}_{j}\,,$
(4)

where the Einstein summation convention is implied over the spinor indices.
Furthermore, we will make use of simple spinor chains, specifically we define

$\displaystyle\begin{split}\langle i|j+k|l]&=\langle ij\rangle[jl]+\langle
ik\rangle[kl]\,,\\\ [i|j+k|l\rangle&=[ij]\langle jl\rangle+[ik]\langle
kl\rangle\,,\end{split}$ (5)

as well as

$\displaystyle\begin{split}\langle i|j+k|l+m|n\rangle&=\langle i|j+k|l]\langle
ln\rangle+\langle i|j+k|m]\langle mn\rangle\,,\\\
[i|j+k|l+m|n]&=[i|j+k|l\rangle[ln]+[i|j+k|m\rangle[mn]\,.\end{split}$ (6)

For brevity, we will often denote an $n\text{-point}$ phase-space point as
$(\lambda,\tilde{\lambda})$.

Physical functions of spinor variables satisfy a number of properties.
Specifically, they have well-defined mass dimension and little-group weights.
Working with some function $\mathcal{E}(\lambda,\tilde{\lambda})$ with well-
defined mass dimension means that if we uniformly scale all of the spinors by
$z$ then we find that

$\mathcal{E}(z\lambda_{1},z\lambda_{2},\ldots,z\tilde{\lambda}_{1},z\tilde{\lambda}_{2},\ldots)=z^{2[\mathcal{E}]}\mathcal{E}(\lambda_{1},\lambda_{2},\ldots,\tilde{\lambda}_{1},\tilde{\lambda}_{2}\ldots)\,,$
(7)

where $[\mathcal{E}]$ is the mass dimension of $\mathcal{E}$. If $\mathcal{E}$
has well-defined little-group weight $k$ then this means that if we scale
$\lambda_{k}$ by $z$ and $\tilde{\lambda}_{k}$ by $1/z$ we find

$\mathcal{E}(\ldots,z\lambda_{k},\ldots,\tilde{\lambda}_{k}/z,\ldots)=z^{\\{\mathcal{E}\\}_{k}}\mathcal{E}(\ldots,\lambda_{k},\ldots,\tilde{\lambda}_{k},\ldots)\,,$
(8)

where $\\{\mathcal{E}\\}_{k}$ is the $k^{\text{th}}$ little-group weight of
$\mathcal{E}$. Importantly, this rescaling does not affect the validity of Eq.
(2). We will always work with functions $\mathcal{E}$ with well-defined mass
dimension and little-group weights. Finally, note that for such functions, it
can be useful to compute the mass-dimension and little-group weights
numerically via Eqs. (7) and (8).

Scattering amplitudes have well-defined mass dimension and little-group
weights. The mass dimension depends only on the multiplicity of the process.
For an $n\text{-point}$ amplitude $\mathcal{A}_{n}$ it is well known that the
mass dimension is given by

$[\mathcal{A}_{n}]=4-n\,.$ (9)

The little-group weights depend on the helicity states111As standard, massless
$s\text{-spin}$ states will have helicity $\pm s$, i.e. $h=0$, $\pm 1/2$ and
$\pm 1$ for scalars, spin-1/2 fermions and vectors respectively. of the
scattered particles, more precisely the $k^{\text{th}}$ little-group weight
depends on the helicity state of the $k^{\text{th}}$ particle, denoted as
$h_{k}$. In the all-outgoing convention, the $k^{\text{th}}$ little-group
weight is given by

$\\{\mathcal{A}_{n}\\}_{k}=-2\,h_{k}\,.$ (10)

Beyond tree level, when working in dimensional regularization, a scattering
amplitude can be decomposed as a linear combination of so-called “master
integrals”. Such a decomposition can be written as

$\mathcal{A}^{(l)}_{n}=\sum_{i}\mathcal{B}_{i}(\lambda,\tilde{\lambda},\epsilon)\mathcal{I}_{i}(\lambda,\tilde{\lambda},\epsilon)\,,$
(11)

where the $\mathcal{B}_{i}$ are rational functions of the spinors
$(\lambda,\tilde{\lambda})$ and of the dimensional regulator $\epsilon$, and
the $\mathcal{I}_{i}$ are transcendental functions thereof. It is well
understood that amplitudes in gauge theory diverge in a universal way
Catani:1998bh ; Becher:2009cu ; Gardi:2009qi ; Becher:2009qa , (see Ref.
Agarwal:2021ais  for a recent review). Specifically, after renormalization,
these divergences can be written in terms of lower loop amplitudes and
universal operators. That is, one can write

$\mathcal{A}^{(l)}_{n,R}=\sum_{l^{\prime}=0}^{l-1}{\bf
I}^{(l-l^{\prime})}\mathcal{A}^{(l^{\prime})}_{n,R}+\mathcal{H}_{n}^{(l)}+\mathcal{O}(\epsilon)\,,$
(12)

where $\mathcal{A}^{(l)}_{n,R}$ is the renormalized $l-\text{loop}$ amplitude
and we have introduced the so-called “finite remainder”
$\mathcal{H}_{n}^{(l)}$, which captures the new information at each
perturbative order. In practice, one computes the finite remainder by
inserting the $\epsilon$ expansion of the master integrals into Eq. (11) and
subtracting the lower loop contributions in Eq. (12). The resulting expression
for the finite remainder can be expressed in a basis of special functions.
That is, in general we can write

$\mathcal{H}^{(l)}_{n}=\sum_{i}\mathcal{C}_{i}(\lambda,\tilde{\lambda})\mathcal{F}_{i}(\lambda,\tilde{\lambda})\,,$
(13)

where the $\mathcal{C}_{i}$ are rational functions of the spinors and the
$\mathcal{F}_{i}$ are special functions of the spinors. In this work, we
consider the $\mathcal{C}_{i}$ in common denominator form. Specifically, we
write

$\mathcal{C}_{i}(\lambda,\tilde{\lambda})=\frac{\mathcal{N}_{i}(\lambda,\tilde{\lambda})}{\prod_{j=1}^{n_{i}}\mathcal{D}_{j}(\lambda,\tilde{\lambda})^{q_{ij}}}\,,$
(14)

where $n_{i}$ is the number of denominator factors and the $\mathcal{N}_{i}$
and $\mathcal{D}_{j}$ are polynomials of spinors. As is well known, the
amplitude only picks up a little-group rescaling under Lorentz transformations
and so it can only depend on the spinors indirectly through the spinor
brackets of Eq. (4). In practice, it is trivial to choose the basis of
transcendental functions $\mathcal{F}_{i}$ to also have this property, and so
the coefficient functions inherit it as well. Importantly, the
$\mathcal{N}_{i}$ and $\mathcal{D}_{j}$ all have well-defined mass dimension
and little-group weights.

For the rest of this work, we shall work in a framework where we are able to
numerically evaluate the $\mathcal{C}_{i}$ over an arbitrary field. In
practice, this may be when one has an explicit analytic form available, or an
appropriate algorithm to compute the $\mathcal{C}_{i}$. Our aim is then to use
this numerical information in an efficient way to determine the analytic form
of the functions $\mathcal{C}_{i}$.

###  2.1 Rudiments of Algebraic Geometry

The coefficients $\mathcal{C}_{i}$ in a scattering amplitude have been
introduced in the previous section as ratios of polynomials, which are to be
evaluated on inputs that satisfy momentum-conservation relations. In this
section, we introduce basic technologies of algebraic geometry which will
allow us to understand polynomials in this context in detail. We intend our
presentation to be self-contained and we refer the reader to Refs.
cox1994ideals ; Zhang:2016kfo ; becker2012groebner  for an introductory
account of the requisite algebraic geometry.

####  2.1.1 Polynomials, Ideals and Varieties

The central object of study will be polynomials in spinor variables.
Therefore, we consider the polynomial ring of spinor variables for $n$
massless particles,

$S_{n}=\mathbb{F}\bigl{[}\lambda_{10},\lambda_{11},\ldots,\lambda_{n0},\lambda_{n1},\tilde{\lambda}_{1{\dot{0}}},\tilde{\lambda}_{1{\dot{1}}},\ldots,\tilde{\lambda}_{n{\dot{0}}},\tilde{\lambda}_{n{\dot{1}}}\bigr{]}\,,$
(15)

where $\mathbb{F}$ is the coefficient field and the variables are the various
$\lambda_{i\alpha}$ and $\tilde{\lambda}_{j\dot{\alpha}}$. All polynomials in
spinor variables are elements of $S_{n}$. For example, the spinor brackets
defined in Eq. (4) can be identified as elements of $S_{n}$. Furthermore,
constraints on the spinors, such as momentum conservation (3) or being on a
particular surface, are expressed using elements of $S_{n}$. Here, and
throughout this work, we shall abstract over the field $\mathbb{F}$. In
practice, we can consider $\mathbb{F}$ to be the rational numbers
$\mathbb{Q}$, the real numbers $\mathbb{R}$, the complex numbers $\mathbb{C}$,
a finite field $\mathbb{F}_{p}$ or the $p\kern 0.80002pt\text{-adic}$ numbers
$\mathbb{Q}_{p}$, which we will discuss in Section 3.1. For theoretical
considerations, such as considering the geometry, we will always work in an
algebraically closed field such as the complex numbers. For practical
calculations, we will be working over finite fields or $p\kern
0.80002pt\text{-adic}$ numbers. Throughout this work, we will take this
polynomial perspective as our foundation. This perspective explicitly breaks
Lorentz covariance in intermediate stages of our calculation, as we work in a
given frame. Furthermore, the ring $S_{n}$ contains unphysical polynomials,
such as ones without well-defined mass dimension and little-group weight. We
will return to the question of imposing these constraints in Section 2.2.

##### Ideals.

The key algebraic object that we use is a so-called ideal. We will work with
rings, such as polynomial rings, in which ideals are finitely generated.
Specifically, we consider a set of elements $\\{p_{1},\ldots,p_{k}\\}\in A$
called generators and define an ideal of $A$ as

$\bigl{\langle}p_{1},\ldots,p_{k}\bigr{\rangle}_{A}\,=\,\left\\{\sum_{i=1}^{k}a_{i}p_{i},\,\,a_{i}\in
A\right\\}.$ (16)

Here, as we will work in a number of rings, we extend the
$\langle\ldots\rangle$ notation of Ref. cox1994ideals  with a subscript to
denote the ring under consideration. When discussing an ideal we typically
label it as $J$ or $K$. From the definition in Eq. (16), it is clear that an
ideal always forms a subset of the ring $A$. In the case where the subset is
proper, that is when we have an ideal $J$ such that $J\subsetneq A$, we say
that $J$ is a proper ideal. We refer to the set $\\{p_{1},\ldots,p_{k}\\}$ as
a generating set of the ideal. In practice we will always consider physical
generating sets, i.e. sets in which each $p_{i}$ has well-defined mass
dimension and little-group weight. Furthermore, we will often consider ideals
that are generated by multiple elements which can be grouped into an object
with some open spinor index. In this case we will use a natural shorthand
where we do not write the individual generators, but only the object with an
open index. As a simple example consider

$\langle\lambda_{j\alpha}\rangle_{S_{n}}\overset{!}{=}\langle\lambda_{j0},\lambda_{j1}\rangle_{S_{n}}\,.$
(17)

Multiple generating sets may correspond to the same ideal, i.e. they are not
unique, and different generating sets of the same ideal may have a different
number of elements. For any ideal $J$, there exist generating sets with a
minimal number of elements and any such generating set is called a minimal
generating set. The size of a minimal generating set is denoted by

$\mu(J)=\mathrm{min}\left(\big{\\{}|G|\,\,:\,\,G\text{ is a generating set of
}J\big{\\}}\right),$ (18)

where $|G|$ denotes the number of elements of the generating set $G$. While
still not unique, we will always present ideals through minimal generating
sets. For ideals generated by homogeneous polynomials, such as those which we
consider, minimal generating sets can be determined algorithmically222For
example, one can find such an algorithm implemented in the computer algebra
system Singular DGPS  under the minbase command.. In practice, we will always
have a generating set of the ideal at hand. A trivial example of an ideal is
the ideal generated by the zero element of the ring. This is the set
containing only the zero element, that is

$\langle 0\rangle_{A}=\\{0\\}\,.$ (19)

As algebraic objects, ideals have natural algebraic operations associated to
them. For example, we will make use of the ideal sum, which we define through

$\langle p_{1},\ldots,p_{k}\rangle_{A}+\langle
q_{1},\ldots,q_{l}\rangle_{A}=\langle
p_{1},\ldots,p_{k},q_{1},\ldots,q_{l}\rangle_{A}\,.$ (20)

Furthermore, one can take the ideal product. Given two ideals $J=\langle
p_{1},\ldots,p_{a}\rangle_{A}$ and $K=\langle q_{1},\ldots,q_{b}\rangle_{A}$,
we define the ideal product $JK$ as

$JK=\langle p_{i}q_{j}\,\,:\,\,1<i\leq a\,,\;1<j\leq b\rangle_{A}\,,$ (21)

that is, the generators of $JK$ are the products of the generators of $J$ and
$K$. It is clear that the ideal product is commutative, i.e. $JK=KJ$. It will
also be useful to consider the ideal power $J^{k}$, which we define
recursively through

$J^{0}=\langle 1\rangle_{A}\quad\;\text{and}\quad\;J^{k}=JJ^{k-1}\,.$ (22)

When working with an ideal $J$ in $S_{n}$, we will often be interested in
other ideals which can be constructed from $J$ by parity or permutations of
the associated spinors. We define a permuted ideal through

$J(\sigma(1)\dots\sigma(n))=J|_{\lambda_{i}\rightarrow\lambda_{\sigma(i)},\,\tilde{\lambda}_{i}\rightarrow\tilde{\lambda}_{\sigma(i)}}\,,$
(23)

where $\sigma$ is a permutation of $\\{1,\ldots,n\\}$. We will also consider
the parity conjugate ideal $\overline{J}$ defined by a swap of the $\lambda$
and $\tilde{\lambda}$ spinors, that is

$\overline{J}=J|_{\lambda_{\alpha}\leftrightarrow\tilde{\lambda}_{\dot{\alpha}}}\,.$
(24)

In practice, one computes generating sets of the these ideals by applying the
permutation/parity conjugation to the generators of $J$.

##### Algebraic Varieties.

Now, note that for $n\text{-point}$ spinor space, we can regard the tuple of
spinor variables
$\\{\lambda_{10},\lambda_{11},\ldots,\lambda_{n0},\lambda_{n1},\tilde{\lambda}_{1{\dot{0}}},\tilde{\lambda}_{1{\dot{1}}},\ldots,\tilde{\lambda}_{n{\dot{0}}},\tilde{\lambda}_{n{\dot{1}}}\\}$
as taking values in the $4n\text{-dimensional}$ space $\mathbb{F}^{4n}$.
Physical spinors are constrained to satisfy momentum conservation according to
Eq. (3). It is therefore natural to consider the set of solutions of momentum
conservation in $\mathbb{F}^{4n}$, which defines a so-called algebraic
variety. In general, we can associate a variety to any ideal in $S_{n}$. That
is, given an ideal $J=\langle p_{1},\ldots,p_{k}\rangle_{S_{n}}$, the
associated algebraic variety is defined as

$V(\langle
p_{1},\ldots,p_{k}\rangle_{S_{n}})=\Big{\\{}(\lambda,\tilde{\lambda})\in\mathbb{F}^{4n}\,:\,p_{i}(\lambda,\tilde{\lambda})=0\,\,\mathrm{for}\,\,1\leq
i\leq k\Big{\\}}\,.$ (25)

From this definition it is clear that $V(J)\subseteq\mathbb{F}^{4n}$ for any
ideal $J$. While we have defined an algebraic variety over an arbitrary field
$\mathbb{F}$, many powerful theorems of algebraic geometry can be applied only
when $\mathbb{F}$ is an algebraically closed field, such as the complex
numbers. In this paper, we will always work over these fields when considering
geometry. We remark that the definition of a variety in Eq. (25) allows
varieties to be “reducible”, an important fact we shall return to in detail in
Section 2.3. We note two trivial cases: $V(\langle 1\rangle_{S_{n}})$
corresponds to the empty variety and $V(\langle 0\rangle_{S_{n}})$ corresponds
to all of $\mathbb{F}^{4n}$.

In the same way that we have just associated a variety to an ideal, we can
naturally associate an ideal to a variety. Specifically, for a variety $U$ in
$\mathbb{F}^{4n}$ it turns out that the set of polynomials that vanish on $U$
forms an ideal, which is defined as

$I(U)=\Big{\\{}p\in S_{n}\,:\,p(\lambda,\tilde{\lambda})=0\,\,\text{for
all}\,\,(\lambda,\tilde{\lambda})\in U\Big{\\}}\,.$ (26)

##### Application to Momentum Conservation.

To understand these ideas in a physical context, consider the ideal of $S_{n}$
generated by the four momentum-conservation polynomials of Eq. (3), which we
denote as

$J_{\Lambda_{n}}=\left\langle\sum_{i=1}^{n}\lambda_{i\alpha}\tilde{\lambda}_{i\dot{\alpha}}\right\rangle_{S_{n}}.$
(27)

We will refer to $J_{\Lambda_{n}}$ as “the momentum-conservation ideal”.
Physically, $J_{\Lambda_{n}}$ is the set of all polynomials in spinor
variables which are rewritings of zero. The associated variety is the set of
points in spinor space that satisfy momentum conservation. We dub this the
“momentum-conservation variety” and it is denoted as $V(J_{\Lambda_{n}})$. All
varieties of interest in this work will be sub-varieties of
$V(J_{\Lambda_{n}})$, as physical configurations of spinors must satisfy
momentum conservation. We note that

$J_{\Lambda_{n}}=I(V(J_{\Lambda_{n}}))\,.$ (28)

That is, the momentum-conservation ideal contains all polynomials which vanish
on the momentum-conservation variety. In an algebraically closed field, the
operation in Eq. (28) of taking the ideal associated to the variety associated
to an ideal corresponds to taking the radical of an ideal (cox1994ideals, ,
Chapter 4), which we will denote as

$\sqrt{J}=I(V(J))\,.$ (29)

If it is the case that $\sqrt{J}=J$, then the ideal $J$ is said to be
“radical” (see Appendix A for the algebraic definition). Therefore, we see
that Eq. (28) says that $J_{\Lambda_{n}}$ is radical.

####  2.1.2 Independent Sets and Dimension

A natural question to ask of any geometric structure is its dimension.
Varieties, as surfaces defined by algebraic equations, indeed have a concept
of dimension that we can associate to them. Furthermore, one can also
associate a concept of dimension to an ideal. In this section, we introduce
the concepts relevant for our work and refer the reader to Ref.
(becker2012groebner, , Section 6.3) for a deeper treatment.

In order to ease the discussion, we will work over the polynomial ring
$\mathbb{F}[X_{1},\ldots,X_{n}]$. We will denote the collection of variables
as $\underline{X}=\\{X_{1},\dots,X_{n}\\}$. We will further denote a subset of
the variables as $\underline{Y}\subseteq\underline{X}$. To begin phrasing the
question of dimension we ask if the variables $\underline{Y}$ can be chosen
independently on the variety $V(J)$. The important observation is that the
answer will be ‘no’ if there is some polynomial in the ideal $J$ which depends
only on the variables $\underline{Y}$. If there is no such polynomial, then
the variables are not constrained in terms of each other, and so the variables
$\underline{Y}$ can be chosen independently. We can formally state the
question of the existence of such a polynomial by considering the associated
elimination ideal defined as333This can be computed via Gröbner basis methods,
see e.g. Ref. (cox1994ideals, , Chapter 3). Nevertheless, computation of the
elimination ideal can be avoided when computing the dimension.

$J_{\underline{Y}}=J\cap F[\underline{Y}]\,,$ (30)

that is, the intersection of the ideal $J$ with the set of all polynomials
which depend only on the variables $\underline{Y}$. An independent set
$\underline{Y}$ of a proper ideal $J$ is defined by requiring that the
associated elimination ideal $J_{\underline{Y}}$ contains only the zero
element, that is

$J_{\underline{Y}}=\\{0\\}\quad\Rightarrow\quad\underline{Y}\text{ is an
independent set of }J\,.$ (31)

Furthermore, an independent set is said to be maximally independent if there
exists no other independent set which contains it. That is,

$\underline{Y}\text{ is maximally independent if
}\nexists\;\,\underline{Y^{\prime}}\supset\underline{Y}\,,\,\text{ with
}\underline{Y^{\prime}}\text{ and }\underline{Y}\text{ independent sets}\,.$
(32)

Note that, in the general case, not all maximally independent sets need be of
the same length.

With the definition of independent sets in hand, we can now discuss dimension.
Specifically, for a proper ideal $J$ of a polynomial ring
$\mathbb{F}[\underline{X}]$, the dimension of $J$ is defined as

$\dim(J)=\text{max}\left(\\{|\underline{Y}|\,\,:\,\,\underline{Y}\,\text{ is
an independent set of }\,J\\}\right)\,.$ (33)

That is, the dimension of the ideal is the length of the largest independent
set. It is clear that this length is unique, as there will always exist at
least one independent set (the empty set) and we take the length of the
largest independent set. The dimension of the variety associated to $J$ is
defined as

$\dim\big{(}V(J)\big{)}=\dim(J)\,.$ (34)

To build intuition, consider the trivial ideal $\langle 0\rangle_{A}$. In the
case where $A$ is a polynomial ring, as there are no constraints,
$\dim\big{(}\langle 0\rangle_{A}\big{)}$ naturally coincides with the number
of variables. As a second example, consider a case where $V(J)$ corresponds to
a finite set of points, then all variables are fixed on each point and so
there is no non-empty independent set. One thus finds that $\dim(J)=0$.
Naturally, $J$ is called a zero-dimensional ideal. Importantly, efficient
algorithms exist to compute both the maximally independent sets of an ideal
$J$ and $\mathrm{dim}(J)$ given a Gröbner basis of $J$, see e.g. Ref.
(becker2012groebner, , Proposition 9.29)444Implementations of algorithms to
compute maximally independent sets and dimensions of ideals can be found in
computer algebra systems such as Singular..

Finally, we introduce the notion of codimension of an ideal. Specifically, for
a proper ideal $J$ in a ring $A$, we define

$\mathrm{codim}(J)=\dim(\langle 0\rangle_{A})-\dim(J)\,.$ (35)

With this language then we see that

$\mathrm{codim}(J_{\Lambda_{n}})=4\,.$ (36)

Intuitively, this can be regarded as the number of constraints imposed by the
generators of an ideal. We remark that the number of constraints may be less
than the number of generators. If the codimension of an ideal is equal to the
length of its minimal generating set, we say that this ideal has maximal
codimension, that is

$\mu(J)=\text{codim}(J)\,\,\,\Rightarrow\,\,\,J\text{ is of maximal
codimension.}$ (37)

For example, the momentum-conservation ideal $J_{\Lambda_{n}}$ is of maximal
codimension.

####  2.1.3 Gröbner Bases

In order to make practical use of the concepts that we present in this paper,
two major tools are polynomial reduction and Gröbner bases. Here we review
these objects in order to set up notation but, as these are common tools in
the particle physics literature, we refer the reader to Ref. cox1994ideals
for a pedagogical introduction. As we work in a number of polynomial rings, in
this section we shall maintain the generic notation introduced in the previous
section. A polynomial ring $\mathbb{F}[\underline{X}]$, with
$\underline{X}=\\{X_{1},\ldots,X_{n}\\}$, can be viewed as a (countably)
infinite-dimensional vector space – the direct sum of one-dimensional spaces
corresponding to the monomials, which we denote as

$\underline{X}^{\underline{\alpha}}=\prod_{i=1}^{n}X_{i}^{\alpha_{i}}\,,$ (38)

where $\underline{\alpha}\in\mathbb{Z}_{\geq 0}^{n}$. That is, each
$\alpha_{i}$ is a non-negative integer. A polynomial $p$ in
$\mathbb{F}(\underline{X})$ takes the form

$p=\sum_{\underline{\alpha}\in\mathbb{Z}_{\geq
0}^{n}}c_{\underline{\alpha}}\underline{X}^{\underline{\alpha}},\qquad
c_{\underline{\alpha}}\in\mathbb{F}\,,$ (39)

where only a finite number of $c_{\underline{\alpha}}$ are non-zero. A useful
structure to put on the space is a so-called monomial ordering, denoted by
$\succeq$. This is a (total) ordering of the exponents $\underline{\alpha}$ of
the monomials. Common orderings are “lexicographic” and “degree reverse
lexicographic”. Both of these orderings depend on an underlying ordering of
the variables $\underline{X}$. Unless otherwise stated, throughout this work
we use the degree reverse lexicographic ordering. Given an ordering $\succeq$,
one can organize the terms of any polynomial and thereby define a lead
monomial, given by

$\mathrm{LM}(p)=\underline{X}^{\underline{\beta}}\,,\;\,\text{where}\;\;\underline{\beta}=\mathrm{max}_{\succeq}\left(\\{\underline{\alpha}:c_{\underline{\alpha}}\neq
0\\}\right)\,,$ (40)

where the maximum is taken over the set with respect to the ordering
$\succeq$. An important application of the lead monomial of a polynomial is to
define the concept of reducibility of one polynomial by another. Specifically,
one says that $p$ is reducible by $h$ if the lead monomial of $h$ is a factor
of the lead monomial of $p$, that is

$\mathrm{LM}(p)\,\,|\,\,\mathrm{LM}(h)\quad\Rightarrow\quad p\text{ is
reducible by }h\,,$ (41)

where we use $x\,|\,y$ to denote that $y$ is a factor of $x$. If $y$ does not
factor $x$ we write $x\nmid y$. If the lead monomial of $h$ is not a factor of
the lead monomial of $p$ then we say that $p$ is irreducible by $h$. Given a
set of generators $H=\\{h_{1},\ldots,h_{k}\\}$ of an ideal $J$, one can then
discuss polynomial reduction. Specifically, it turns out that one can always
write a polynomial $p$ as

$p=q_{1}h_{1}+\ldots+q_{k}h_{k}+\Delta_{H}(p)\,,$ (42)

where $\Delta_{H}(p)$ is irreducible by any of the $h_{i}$. The object
$\Delta_{H}(p)$ is of fundamental importance and is known as the remainder
modulo $H$. An important feature of remainders is that they are linear
combinations of monomials that are irreducible by the given generating set,
that is

$\Delta_{H}(p)=\sum_{{\underline{\beta}}\in\mathrm{irreds}(H)}d_{\underline{\beta}}\underline{X}^{\underline{\beta}}\,,\;\,\text{where}\;\,\mathrm{irreds}(H)=\\{\underline{\beta}\;:\;\underline{X}^{\underline{\beta}}\,\,\nmid\mathrm{LM}(h)\;\forall\;h\,\in\,H\\}\,.$
(43)

Here $\mathrm{irreds}(H)$ is the set of exponents whose associated monomial is
not a (polynomial) multiple of the lead monomial of any element of the set
$H$. The aim of introducing the remainder $\Delta_{H}(p)$ is to define a
canonical form of $p$ when working modulo elements of the ideal $J$. However,
it turns out that the remainder modulo $H$ is not uniquely determined by the
ordering $\succeq$ and the ideal $J$ that it generates. It also depends on the
details of the set $H$. Specifically, if a polynomial is reducible by an
element of the ideal $J$ it may not be reducible by an element of the
generating set $H$. However, given an ordering $\succeq$, there exist special
generating sets of $J$ that do uniquely determine the remainder. These are
known as Gröbner bases. We denote a Gröbner basis of an ideal $J$ as
$\mathcal{G}(J)$. General algorithms exist to compute Gröbner bases, which are
implemented in many computer algebra systems. Remainders modulo a Gröbner
basis have the important property that if $p$ is in the ideal, then the
remainder is zero. That is,

$p\in J\quad\Leftrightarrow\quad\Delta_{\mathcal{G}(J)}(p)=0\,.$ (44)

##### Organizing Vector Spaces by Ideals.

A useful application of Gröbner bases is to split a subspace of a polynomial
ring into a subspace that belongs to an ideal, and a remaining subspace.
Specifically, consider the polynomial ring $\mathbb{F}[\underline{X}]$, an
ideal $J$ of $\mathbb{F}[\underline{X}]$ and a finite-dimensional vector space
$W$ that is a subspace of $\mathbb{F}[\underline{X}]$. Using $J$, one can
split the space $W$ into a direct sum as

$W\cong(W\cap J)\oplus W/(W\cap J)\,.$ (45)

The left summand $(W\cap J)$ is the subspace of $W$ formed by all elements
that are also elements of $J$. The right summand $W/(W\cap J)$ is the quotient
of $W$ by this subspace. $W/(W\cap J)$ can be considered as the space of
elements of $W$ modulo the elements of $J$. To make practical use of the
decomposition in Eq. (45), we now discuss how to find a basis of the two
summand spaces.

Let us first consider how to find a basis of $W\cap J$ given a basis
$\Omega=\\{\Omega_{1},\ldots,\Omega_{\dim(W)}\\}$ of $W$. We recall from Eq.
(44) that all elements of $J$ have zero remainder modulo $\mathcal{G}(J)$. As
polynomial division acts linearly on $W$, we consider the remainders of the
basis elements, $\Delta_{\mathcal{G}(J)}(\Omega_{j})$. Recall that these
remainders can be expressed in terms of monomials irreducible by
$\mathcal{G}(J)$, that is

$\Delta_{\mathcal{G}(J)}(\Omega_{j})=\sum_{\underline{\beta_{i}}\in\mathrm{irreds}(\mathcal{G}[J])}\Delta_{ij}\\!\left({\mathcal{G}[J]},\Omega\right)\underline{X}^{\underline{\beta_{i}}}\,,$
(46)

where $\Delta_{ij}\\!\left({\mathcal{G}[J]},\Omega\right)$ is the
$\mathbb{F}\text{-valued}$ matrix of coefficients of the remainder of
$\Omega_{j}$ when expressed in terms of the monomials
$\underline{X}^{\underline{\beta_{i}}}$. Note that for ideals that are not
zero dimensional, $\mathrm{irreds}(\mathcal{G}[J])$ is an infinite set.
However, as in practice the degree of $\Omega_{j}$ is bounded, the sum is
always finite. Clearly, one can linearly express the remainder of any element
of $W$ in terms of the $\Delta_{\mathcal{G}(J)}(\Omega_{j})$. Therefore, we
see that any element $w$ of $W\cap J$ takes the form

$w=\sum_{j=1}^{\dim(W)}c_{j}\Omega_{j},\quad\text{such
that}\quad\sum_{j=1}^{\dim(W)}\Delta_{ij}\\!\left({\mathcal{G}[J]},\Omega\right)c_{j}=0\,,$
(47)

where $c_{j}\in\mathbb{F}$. Eq. (47) states that the $c_{j}$ live in the
nullspace of the $\mathbb{F}\text{-valued}$ matrix
$\Delta_{ij}\\!\left({\mathcal{G}[J]},\Omega\right)$. A basis of this
nullspace can be computed with standard linear algebra techniques. Through Eq.
(47), we then arrive at a basis of $W\cap J$.

Next, we consider how to construct a set of elements of $W$ that form a basis
of $W/(W\cap J)$ when considered modulo elements of $W\cap J$. Specifically,
we show that one can choose a subset of the basis elements of $W$ using
standard linear-algebra techniques. To see this, note that $W/(W\cap J)$ is
isomorphic to the space spanned by the remainders modulo $\mathcal{G}(J)$ of
the elements of $W$. Furthermore, considering $i$ as a row index and $j$ as a
column index, this space is isomorphic to the column space of the matrix
$\Delta_{ij}\\!\left({\mathcal{G}[J]},\Omega\right)$. We then see that

$W/(W\cap
J)\,\,\cong\,\,\mathrm{span}_{\mathbb{F}}\left(\Big{\\{}\Omega_{j}\quad\text{such
that}\quad{j\in\mathrm{pivots}\left[\Delta_{ij}\\!\left({\mathcal{G}[J]},\Omega\right)\right]}\Big{\\}}\right)\,.$
(48)

That is, a basis of $W/(W\cap J)$ can be chosen as the subset of $\Omega$
corresponding to the pivot columns of the matrix
$\Delta_{ij}\\!\left({\mathcal{G}[J]},\Omega\right)$. Note that constructing
$W/(W\cap J)$ in this way gives a true subspace of $W$, rather than one up to
isomorphism. Therefore, with the construction in Eq. (48), Eq. (45) is an
equality. To compute the set of pivot indices, one can use the standard
technique where the pivot indices are read from the row-reduced echelon form
of $\Delta_{ij}\\!\left({\mathcal{G}[J]},\Omega\right)$. We note that the
subset of $\Omega$ that is chosen as a basis by this algorithm depends on the
ordering of the elements of $\Omega$. Specifically, elements of $\Omega$ that
occur earlier in the set are prioritized.

####  2.1.4 Quotient Rings

When considering physical polynomials in spinor variables, i.e. polynomials
subject to momentum conservation, it is easy to see that the polynomial ring
$S_{n}$ is redundant. Specifically, we wish to consider a number of elements
of $S_{n}$ as equivalent: those which can be converted into each other by
application of the momentum-conservation identity. In numerical applications,
where one only has access to evaluations of functions on points
$(\lambda,\tilde{\lambda})\in V(J_{\Lambda_{n}})$, this is essential: any two
polynomials $p,q\in S_{n}$ that are equivalent under momentum conservation
will evaluate to the same value on such a point. Hence, the momentum-
conservation ideal induces an equivalence class of polynomials: we wish to
consider two polynomials in $S_{n}$ which can differ by some element of
$J_{\Lambda_{n}}$ as equivalent. That is, for $p,q\in S_{n}$

$p\sim q\iff p-q\in J_{\Lambda_{n}}\,.$ (49)

Working in a polynomial ring up to equivalence by an ideal means that we work
in a quotient ring. Specifically, all independent spinor polynomials given
momentum conservation belong to the quotient ring

$R_{n}=S_{n}/J_{\Lambda_{n}}\,.$ (50)

Returning to Eq. (49), both $p$ and $q$ belong to the same equivalence class
in $R_{n}$, and we say that $p$ and $q$ are representatives of this
equivalence class. In order to represent elements of quotient rings, one can
make use of Gröbner bases. Specifically, if two elements $p$ and $q$ are
equivalent then their difference belongs to the ideal $J_{\Lambda_{n}}$ and so

$\Delta_{\mathcal{G}(J_{\Lambda_{n}})}(p-q)=0\,.$ (51)

Rearranged, this means that the remainders of $p$ and $q$ are equal.
Therefore, elements of a quotient ring are uniquely (canonically) represented
by their remainders modulo a Gröbner basis. For a recent application of
polynomial quotient rings in other areas of particle physics see Ref.
Henning:2017fpj . In this work, we will refer to $R_{n}$ as the set of
“physically inequivalent” polynomials.

##### Ideals and Varieties in Quotient Rings.

It is clear from the definition in Eq. (16) that we can consider ideals in
polynomial quotient rings. Ideals in polynomial quotient rings are also
finitely generated and one can represent generators by a representative of the
equivalence class. Let us denote a polynomial ring by $A$, an ideal of $A$ by
$J$ and consider the polynomial quotient ring $A/J$. It turns out that, for
computations involving an ideal $K$ of $A/J$, we can perform the computation
using an ideal in the polynomial ring $A$ which corresponds to $K$. Therefore,
we are able to continue to use Gröbner basis technology when working with
polynomial quotient rings. Let us concretize the correspondence as follows.
Let us denote the generators of $J$ as $\\{q_{1},\ldots,q_{l}\\}$. We
introduce a map $\pi_{A,A/J}$ which takes an ideal of $A$ to an ideal of the
quotient ring $A/J$. Explicitly we have

$\pi_{A,A/J}\left(\,\left\langle
p_{1},\ldots,p_{k}\right\rangle_{A}\,\right)=\langle
p_{1},\ldots,p_{k}\rangle_{A/J}\,,$ (52)

where the $p_{i}$ on the right hand side are understood as representatives of
the equivalence class. The map $\pi_{A,A/J}$ is many-to-one, but if one
restricts the domain to ideals $K$ of $A$ which contain $J$, then the map is
one-to-one (becker2012groebner, , Lemma 1.63). Given this restriction, the
$\pi_{A,A/J}$ has a unique inverse given by

$\pi^{-1}_{A,A/J}\left(\,\langle
p_{1},\ldots,p_{k}\rangle_{A/J}\,\right)=\left\langle
p_{1},\ldots,p_{k},q_{1},\ldots,q_{l}\right\rangle_{A}\,.$ (53)

Consider $R_{n}$, this means that, given an ideal $J$ of $R_{n}$, one appends
the generators of $J_{\Lambda_{n}}$ to find the corresponding ideal in
$S_{n}$. The map $\pi_{A,A/J}$ has a number of applications. For instance,
given a representative $q$ of an element of $A/J$ and an ideal $K$ of $A/J$,
we define the remainder of $q$ modulo a Gröbner basis of $K$ through

$\Delta_{\mathcal{G}(K)}(q)=\Delta_{\mathcal{G}(\pi^{-1}_{A,A/J}[K])}(q)\,,$
(54)

where the right hand side is again a representative of an element of $A/J$.
This allows us to apply the vector space organization technology of Section
2.1.3 also in quotient rings. Moreover, $\pi_{A,A/J}$ induces a definition of
dimension of ideals in the quotient ring. Specifically, we define

$\dim\left(\langle p_{1},\ldots,p_{k}\rangle_{A/\langle
q_{1},\ldots,q_{l}\rangle_{A}}\right)=\dim\left(\left\langle
p_{1},\ldots,p_{k},q_{1},\ldots,q_{l}\right\rangle_{A}\right)\,.$ (55)

Recalling the definition of codimension in Eq. (35) and the codimension of
momentum conservation in Eq. (36), we see that the dimension of $\langle
0\rangle_{R_{n}}$ is given by

$\mathrm{dim}(\langle 0\rangle_{R_{n}})=4n-4\,.$ (56)

Furthermore, $\pi_{A,A/J}$ gives a natural way to understand the geometry of
ideals in a quotient ring. Specifically, we can apply the correspondence and
consider the variety associated to the associated ideal in $S_{n}$, i.e. we
define

$V\left(\langle
p_{1},\ldots,p_{k}\rangle_{R_{n}}\right)=V\Big{(}\Big{\langle}p_{1},\ldots,p_{k},\sum_{i=1}^{n}\lambda_{i\alpha}\tilde{\lambda}_{i\dot{\alpha}}\,\Big{\rangle}_{S_{n}}\,\Big{)}\,.$
(57)

Therefore, all varieties associated to ideals in $R_{n}$ are sub-varieties of
the of the momentum-conservation variety $V(J_{\Lambda_{n}})$.

Finally, we point out that we will also make use of ideals of $R_{n}$
generated by the application of a permutation or parity operation as denoted
in Eq. (23). Similar to the $S_{n}$ case, one can compute generating sets of
these ideals by applying the permutation and/or parity operation to the
generators in $R_{n}$. This follows as the momentum-conservation ideal
$J_{\Lambda_{n}}$ is invariant under permutations and parity.

###  2.2 Linearly Independent Polynomials in Spinor Brackets

In the previous subsection, we set up an algebro-geometric framework to
understand spinor space. We have seen that all polynomial functions on
$V(J_{\Lambda_{n}})$ are contained in $R_{n}$. In practice, when discussing
scattering amplitudes, we are only interested in a subset of these functions:
those which are Lorentz invariant up to a little-group rescaling. In this
sense, the ring $R_{n}$ is a superset of the polynomial functions relevant for
scattering amplitudes. Furthermore, $R_{n}$ is the set of spinor polynomials
up to equivalences induced by momentum conservation. For the purposes of
making an Ansatz, it is necessary that there are no linear dependencies
between the Ansatz elements. In this section, we discuss how we resolve these
two issues.

##### The Bracket Subring.

Our aim is to understand the set of physically inequivalent polynomial
functions which are Lorentz invariant, up to a little-group rescaling. These
form a subset of $R_{n}$, which we denote by

$\mathcal{R}_{n}=\\{\,a\in
R_{n}\,\,:\,\,\Lambda(a)=Z_{(\Lambda,a)}\,a\,\\}\,,$ (58)

where $\Lambda$ is a Lorentz transformation which is continuously connected to
the identity and $Z_{(\Lambda,a)}$ is an element of $\mathrm{GL}(1)$
corresponding to a little-group rescaling of $a$ when acted on by $\Lambda$.
One can show that $\mathcal{R}_{n}$ is a ring. Therefore, as a subset of
$R_{n}$ which is also a ring, $\mathcal{R}_{n}$ is a subring of $R_{n}$. From
a physical perspective, it is clear that $\mathcal{R}_{n}$ is composed of
polynomials which can be described in terms of spinor brackets. Therefore, we
refer to $\mathcal{R}_{n}$ as the bracket subring. In order to work with
$\mathcal{R}_{n}$ in practice, it is convenient to reformulate it in a way
that manifests the Lorentz transformation properties of its elements. To this
end, we will observe that we can describe $\mathcal{R}_{n}$ as a polynomial
quotient ring. In this formulation, one can then use Gröbner basis technology
for standard operations, such as finding a canonical form of elements of
$\mathcal{R}_{n}$, checking equivalence of elements of $\mathcal{R}_{n}$ and
intersecting a subspace of $\mathcal{R}_{n}$ with an ideal of
$\mathcal{R}_{n}$.

Let us consider a polynomial ring where we label the variables by the
independent spinor brackets for $n$ particles,

$\mathcal{S}_{n}=\mathbb{F}\bigl{[}\,\langle 12\rangle,\langle
13\rangle,\ldots,\langle(n-1)n\rangle,[12],[13],\ldots,[(n-1)n]\bigr{]}\,.$
(59)

We note that $\mathcal{S}_{n}$ is a polynomial ring in $2\binom{n}{2}$
variables, that is we choose our variables to be $\langle ij\rangle$ and
$[ij]$ for $i<j$. We stress that, in the context of $\mathcal{S}_{n}$, the
spinor brackets are to be considered as variables and not as polynomials in
$R_{n}$. It can be shown (see Appendix C) that $\mathcal{R}_{n}$ is isomorphic
to a polynomial quotient ring as

$\mathcal{R}_{n}\cong\mathcal{R}_{n}^{(q)}\,,\quad\text{where}\quad\mathcal{R}_{n}^{(q)}=\mathcal{S}_{n}/(\mathcal{J}_{\Lambda_{n}}+\mathcal{K}_{\Lambda_{n}}+\overline{\mathcal{K}}_{\Lambda_{n}})\,\\\
$ (60)

and

$\displaystyle\mathcal{J}_{\Lambda_{n}}$
$\displaystyle=\Bigg{\langle}{\sum_{\begin{subarray}{c}j=1\\\ j\neq
i,k\end{subarray}}^{n}}\langle ij\rangle[jk]\,\,:\,\,1\leq i\leq n,\,1\leq
k\leq n\Bigg{\rangle}_{\mathcal{S}_{n}}\,,$ (61)
$\displaystyle\mathcal{K}_{\Lambda_{n}}$ $\displaystyle=\big{\langle}\langle
ij\rangle\langle kl\rangle+\langle ik\rangle\langle lj\rangle+\langle
il\rangle\langle jk\rangle\,\,:\,\,1\leq i<j<k<l\leq
n\big{\rangle}_{\mathcal{S}_{n}}\,,$ (62)
$\displaystyle\overline{\mathcal{K}}_{\Lambda_{n}}$
$\displaystyle=\big{\langle}\hskip 2.56073pt[ij][kl]\hskip 2.56073pt+\hskip
2.56073pt[ik][lj]\hskip 2.56073pt+\hskip 2.56073pt[il][jk]\hskip
2.56073pt\,\,:\,\,1\leq i<j<k<l\leq n\big{\rangle}_{\mathcal{S}_{n}}\,.$ (63)

Here, for ease of notation, for brackets with $j\geq i$ we make use of the
identities

$\langle ji\rangle=-\langle ij\rangle\quad\;\text{and}\quad\;[ji]=-[ij]\,.$
(64)

Physically, $\mathcal{J}_{\Lambda_{n}}$ is the set of relations between spinor
brackets generated by the momentum-conservation identities, and
$\mathcal{K}_{\Lambda_{n}}$ and $\overline{\mathcal{K}}_{\Lambda_{n}}$ are the
set of relations generated by the Schouten identities. Eq. (60) says that
$\mathcal{R}_{n}^{(q)}$ is the set of inequivalent spinor bracket polynomials
under this set of identities.

It is natural to ask what happens if we consider of the set of elements of an
ideal $J$ of $R_{n}$ that can be expressed in terms of spinor brackets.
Mathematically, we are inquiring about the object $J\cap\mathcal{R}_{n}$.
Importantly, it can be shown that $J\cap\mathcal{R}_{n}$ is an ideal of
$\mathcal{R}_{n}$. To be able to perform practical computations with
$J\cap\mathcal{R}_{n}$, we wish to find the ideal in $\mathcal{R}_{n}^{(q)}$
to which $J\cap\mathcal{R}_{n}$ maps. Specifically, we need to be able to
construct a generating set of this ideal. To this end, we will make use of the
correspondence between ideals of polynomial rings and polynomial quotient
rings discussed in Section 2.1.4 and begin by working with the polynomial
rings $\mathcal{S}_{n}$ and $S_{n}$. Consider the augmented polynomial ring

$\Sigma_{n}=\mathbb{F}[\langle
12\rangle,\ldots,\langle(n-1)n\rangle,[12],\ldots,[(n-1)n],\lambda_{10},\lambda_{11},\ldots,\tilde{\lambda}_{1\dot{0}},\tilde{\lambda}_{1\dot{1}},\ldots]\,,$
(65)

that is, a polynomial ring whose variables are both the spinor brackets and
the spinor variables. It is clear that $\Sigma_{n}$ contains both
$\mathcal{S}_{n}$ and $S_{n}$ as subrings. Given an ideal $J=\langle
p_{1},\ldots p_{k}\rangle_{S_{n}}$ we construct the ideal

$\kappa[J]=\left\langle p_{1},\ldots,p_{k},\langle
12\rangle-\big{(}\lambda_{10}\lambda_{21}-\lambda_{20}\lambda_{11}\big{)},\ldots,[12]-\big{(}\tilde{\lambda}_{1\dot{0}}\tilde{\lambda}_{2\dot{1}}-\tilde{\lambda}_{2\dot{0}}\tilde{\lambda}_{1\dot{1}}\big{)},\ldots\right\rangle_{\Sigma_{n}}\,.$
(66)

Here, $\kappa[J]$ is generated by the generators of the ideal $J$, as well as
by the relations between the spinor brackets and spinor variables555A very
similar setup can be found in constructing “algebraic dependence relations” in
the Leĭnartas algorithm leinartas1978factorization ; raichev2012leinartas ;
Meyer:2016slj .. It can be shown (see Appendix C) that the set of elements of
$J$ that can be written in terms of spinor brackets correspond to the ideal of
$\mathcal{S}_{n}$ given by

$\kappa[J]\cap\mathcal{S}_{n}\,.$ (67)

This intersection is an example of elimination of variables and can be
computed in practice via Gröbner basis techniques, see e.g. Section 2.4.3 of
Ref. Zhang:2016kfo . We can then use the correspondence map to understand the
ideals of $\mathcal{R}_{n}^{(q)}$ associated to ideals of $R_{n}$. Combining
this with the isomorphism in Eq. (60), for an ideal $J$ of $R_{n}$ we have
that

$J\cap\mathcal{R}_{n}\cong\pi_{\mathcal{S}_{n},\mathcal{R}_{n}^{(q)}}\left(\kappa[\pi_{S_{n},R_{n}}^{-1}(J)]\cap\mathcal{S}_{n}\right)\,,$
(68)

where the right hand side is the ideal in the polynomial quotient ring
formulation of $\mathcal{R}_{n}$. As a first example of this technology, one
can consider the situation where we wish to find the ideal in
$\mathcal{S}_{n}$ corresponding to $\langle 0\rangle_{S_{n}}$. A Gröbner basis
calculation shows that

$\kappa[\langle
0\rangle_{S_{n}}]\cap\mathcal{S}_{n}=\mathcal{K}_{\Lambda_{n}}+\overline{\mathcal{K}}_{\Lambda_{n}}\,,$
(69)

that is, we have only generated the Schouten identities. A less trivial
example is to find the ideal in $\mathcal{S}_{n}$ which corresponds to
$J_{\Lambda_{n}}$. One finds that

$\kappa[J_{\Lambda_{n}}]\cap\mathcal{S}_{n}=\mathcal{J}_{\Lambda_{n}}+\mathcal{K}_{\Lambda_{n}}+\overline{\mathcal{K}}_{\Lambda_{n}}\,.$
(70)

Here, we now pick up both momentum-conservation and Schouten identities.

##### Physical Polynomial Space.

In this work, our aim is to construct compact Ansätze for the rational
prefactors. So far, we have discussed the polynomials relevant for the
numerators of rational prefactors in scattering amplitudes as living in the
spinor bracket ring $\mathcal{R}_{n}$. However, this is an infinite
dimensional vector space, and so this information is insufficient for the
construction of a finite Ansatz. Nevertheless, physical polynomials, such as
numerators of rational prefactors, have well-defined mass dimension and
little-group weight. This leads us to define the space of independent bracket
polynomials with a well-defined mass dimension $d$ and little-group weights
$\phi_{k}$,

$\mathcal{M}_{d,\vec{\phi}}=\Big{\\{}a\in\mathcal{R}_{n}\;:\;[a]=d,\,\mathrm{and}\;\\{a\\}_{k}=\phi_{k}\Big{\\}}\,.$
(71)

Note that as the mass dimension $d$ is fixed, $\mathcal{M}_{d,\vec{\phi}}$ is
a finite-dimensional vector space over $\mathbb{F}$. If we can find a basis of
$\mathcal{M}_{d,\vec{\phi}}$, we can use this basis as an Ansatz for the
numerator polynomial. Furthermore, any Ansatz for the numerator polynomial
must be expressible in terms of a basis of $\mathcal{M}_{d,\vec{\phi}}$.
Therefore, this basis is a natural starting point for refined Ansätze with
special properties. We will now describe an algorithm to construct a basis of
$\mathcal{M}_{d,\vec{\phi}}$. There are two problems we need to solve. First,
we must construct elements of $\mathcal{R}_{n}$ which are linearly
independent. It is clear from the definition of $\mathcal{R}_{n}^{(q)}$ that
monomials in $\mathcal{M}_{d,\vec{\phi}}$ are related by momentum-conservation
and Schouten identities. Second, we must impose the constraints of fixed mass
dimension and little-group weight. We note that other methods have been put
forward to build a basis of $\mathcal{M}_{d,\vec{\phi}}$, see e.g. Ref.
Huber:2021vnc ; DeAngelis:2022qco . Our methods make use of general features
of the algebra of polynomials and we expect them to have wide applicability to
many problems. The approach that we employ here was previously also used in
Ref. Zhang:2012ce , where the problem was finding a linearly independent set
of monomials in loop momentum on a given generalized unitarity cut.

As $\mathcal{R}_{n}$ is isomorphic to a polynomial quotient ring in spinor
brackets, all elements can be expressed as linear combinations of monomials in
the spinor brackets, which we denote as

$m_{(\alpha,\beta)}=\prod_{j=1}^{n}\prod_{i=1}^{j-1}\langle
ij\rangle^{\alpha_{ij}}[ij]^{\beta_{ij}}\,,$ (72)

where $\alpha_{ij}$ and $\beta_{ij}$ belong to $\mathbb{Z}_{\geq 0}$. It will
turn out that we can pick a subset of the monomials in spinor brackets as
basis elements. Specifically, we will show that

$\mathcal{M}_{d,\vec{\phi}}=\mathrm{span}_{\mathbb{F}}\left(M_{d,\vec{\phi}}\right)\quad\text{where
}\quad M_{d,\vec{\phi}}=\left\\{m_{(\alpha,\beta)}\quad\text{such
that}\quad(\alpha,\beta)\in X_{d,\vec{\phi}}\right\\}\,.$ (73)

That is, $\mathcal{M}_{d,\vec{\phi}}$ is the set of all
$\mathbb{F}\text{-linear}$ combinations of the elements of $M_{d,\vec{\phi}}$,
the set of spinor bracket monomials whose exponents lie in the finite set
$X_{d,\vec{\phi}}$. Our task is to determine the set of exponents
$X_{d,\vec{\phi}}$ such that the associated monomials have mass dimension $d$,
little-group weights $\vec{\phi}$ and are linearly independent elements of
$\mathcal{R}_{n}$.

To begin, we discuss the structure of $\mathcal{R}_{n}$ as an
$\mathbb{F}\text{-vector}$ space. As it is a ring, it is an infinite-
dimensional vector space. Physically, a basis of $\mathcal{R}_{n}$ as an
$\mathbb{F}\text{-vector}$ space gives a set of linearly independent
polynomials in spinor brackets when one takes into account the momentum-
conservation and Schouten identities. To resolve these identities, consider a
polynomial quotient ring $A/J$, where $A$ is a polynomial ring over the field
$\mathbb{F}$ and $J$ is an ideal of $A$. We recall from Section 2.1.3 that
elements of $A/J$ can be uniquely expressed as an $\mathbb{F}\text{-linear}$
combination of monomials that are irreducible by the Gröbner basis
$\mathcal{G}(J)$. Therefore, we see that the monomials which are irreducible
by $\mathcal{G}(J)$ form a basis of $A/J$ as an $\mathbb{F}\text{-vector}$
space (cox1994ideals, , Chapter 4.3, Proposition 4). Recall from Eq. (60) that
$\mathcal{R}_{n}$ is isomorphic to the polynomial quotient ring
$\mathcal{R}_{n}^{(q)}=\mathcal{S}_{n}/(\mathcal{J}_{\Lambda_{n}}+\mathcal{K}_{\Lambda_{n}}+\overline{\mathcal{K}}_{\Lambda_{n}})$.
Therefore, viewing $m_{(\alpha,\beta)}$ as an element of
$\mathcal{R}_{n}^{(q)}$, we require that

$m_{(\alpha,\beta)}\nmid\mathrm{LM}(g)\quad\text{for all}\quad
g\in\mathcal{G}(\mathcal{J}_{\Lambda_{n}}+\mathcal{K}_{\Lambda_{n}}+\overline{\mathcal{K}}_{\Lambda_{n}})\,,$
(74)

where
$\mathcal{G}(\mathcal{J}_{\Lambda_{n}}+\mathcal{K}_{\Lambda_{n}}+\overline{\mathcal{K}}_{\Lambda_{n}})$
is the Gröbner basis associated to momentum-conservation and Schouten
identities. Note that the statement that a polynomial is reducible can be
stated as a set of simultaneous linear inequalities. That is, all of the
elements of $(\alpha,\beta)$ must be greater than or equal to the
corresponding entry in the exponent of $\mathrm{LM}(g)$. The irreducibility
constraint, Eq. (74), is the complement of this. In summary, monomials
$m_{(\alpha,\beta)}$ which satisfy the irreducibility constraint form a basis
of $\mathcal{R}_{n}$ as an $\mathbb{F}\text{-vector}$ space.

The constraints of fixed little-group weight and mass dimension translate to
linear constraints on the exponents $(\alpha,\beta)$. First, we consider mass
dimension: all spinor brackets have unit mass dimension, so we can easily
write the mass dimension of a monomial $m_{(\alpha,\beta)}$ as

$[m_{(\alpha,\beta)}]=\sum_{j=1}^{n}\sum_{i=1}^{j-1}\left(\alpha_{ij}+\beta_{ij}\right)=d\,.$
(75)

Next, we consider little-group weight of a monomial $m_{(\alpha,\beta)}$. It
is clear that

$\\{m_{(\alpha,\beta)}\\}_{k}=\sum_{j=1}^{n}\sum_{i=1}^{j-1}\left(\alpha_{ij}\\{\langle
ij\rangle\\}_{k}+\beta_{ij}\\{[ij]\\}_{k}\right)=\phi_{k}\,,$ (76)

where

$\\{\langle
ij\rangle\\}_{k}=\delta_{ik}+\delta_{jk}\qquad\mathrm{and}\qquad\\{[ij]\\}_{k}=-\left(\delta_{ik}+\delta_{jk}\right)$
(77)

are the little-group weights of the angle and square brackets respectively. It
is interesting to note that, as the $\alpha_{ij}$ and $\beta_{kl}$ are non-
negative, Eqs. (75) and (76) cut out a convex polytope.

Together with the irreducibility constraints (74), the constraints of mass
dimension (75) and little-group weights (76) on $(\alpha,\beta)$ define the
set of exponents $X_{d,\vec{\phi}}$. To solve these equations, first note that
the exponents $(\alpha,\beta)$ are non-negative integers. The space cut out by
our equations is bounded, and therefore $X_{d,\vec{\phi}}$ is finite. Solving
these equations is then reduced to enumerating their solutions. Efficient
algorithms to enumerate such non-negative integer solutions are commonly
implemented in computer algebra systems666For example, in the computer algebra
system Mathematica one can simply apply the Solve function, requiring the
solution domain to be the NonNegativeIntegers..

###  2.3 Geometry of Singular Varieties

Let us consider the rational functions $\mathcal{C}_{i}$ from Eq. (14) and in
particular the set of all their possible denominator factors
$\mathcal{D}=\\{\mathcal{D}_{1},\ldots,\mathcal{D}_{n_{i}}\\}$. In general,
this set will depend on the specifics of the considered external kinematics,
together with the definitions of the functions $\mathcal{F}$. As an example of
typical elements of the set $\mathcal{D}$ we can consider all spinor brackets
from Eq. (4). We wish to study the behavior of the rational coefficients when
considered near varieties on which some subset of the denominators
$\mathcal{D}$ vanishes. We dub these varieties “singular varieties”777The term
“singular” here refers to a property of rational functions on the variety, not
to a topological property of the variety itself.. Conventional examples are
configurations where external particles become soft or collinear. We denote a
singular variety as

$U_{\vec{\gamma}}=V\big{(}\langle\mathcal{D}_{\gamma_{1}},\ldots,\mathcal{D}_{\gamma_{m}}\rangle_{R_{n}}\big{)}\,,$
(78)

for some subset of the denominators
$\\{\mathcal{D}_{\gamma_{1}},\ldots,\mathcal{D}_{\gamma_{m}}\\}\subseteq\mathcal{D}$.
We note that, by definition, the variety $U_{\vec{\gamma}}$ corresponds to an
ideal

$J_{\vec{\gamma}}=\langle\mathcal{D}_{\gamma_{1}},\ldots,\mathcal{D}_{\gamma_{m}}\rangle_{R_{n}}\,,$
(79)

which we dub a “singular ideal”. In practice, we will only be considering
ideals generated by one or two denominator factors, i.e. the cases $m=1$ or
$m=2$.

A key feature of the algorithm we present in this paper will be to generate
numerical configurations of spinors which lie close to singular varieties in
order to determine how fast a given rational expression diverges close to the
singular variety. This procedure is complicated by the fact that the singular
varieties may branch and that the degree of divergence may differ close to
different branches of the same variety. Therefore, we will find it necessary
to be able to control which branch we are approaching numerically. The
remainder of this section reviews in general terms the geometric and algebraic
concepts related to branching. We refer the reader to Section 4 of Ref.
cox1994ideals  for more details.

##### Irreducible Varieties.

The key geometric concept related to branching is that of reducibility of a
variety. The object that we wish to consider is that of an irreducible
variety. A variety $U$ is defined to be irreducible if

$U=U_{1}\cup U_{2}\;\;\Rightarrow\;\l U_{1}=U\;\;\text{or}\;\;U_{2}=U\,.$ (80)

In our case, the varieties $U_{\vec{\gamma}}$ in Eq. (78) may well be
reducible. A reducible variety $U$ can be written as a proper union of sub-
varieties and there exists a minimal decomposition

$U=\bigcup_{k=1}^{n_{B}(U)}U_{k}\,,$ (81)

where $n_{B}(U)$ is the number of varieties in the decomposition, each $U_{k}$
is irreducible and $U_{i}\not\subseteq U_{j}$ for all $i\neq j$. We call each
$U_{k}$ a “branch” of $U$ and this last condition is that no branch is
contained within another. A minimal decomposition is unique, up to the order
of the branches $U_{k}$ (cox1994ideals, , Section 4.6, Theorem 4).

Importantly, the reducibility of a variety may not be manifest from the set of
equations used to define it. Our task is now to discuss how to understand the
decomposition of Eq. (81) algebraically so that one can perform the
decomposition systematically.

##### Ideals Associated to Irreducible Varieties.

To understand the decomposition of varieties in an algebraic fashion, let us
start by considering an ideal $J$, its associated variety $U=V(J)$ and its
minimal decomposition $U=\bigcup_{k}^{n_{B}(U)}U_{k}$. It is natural to
consider the ideal associated to each branch $U_{k}$

$P_{k}=I(U_{k})\,.$ (82)

Here we have judiciously labeled the ideal as $P$, which hints at the fact
that the ideal associated to an irreducible variety is prime (see Appendix A
for the algebraic definition).

A reasonable expectation could be to express $J$ in terms of the $P_{k}$’s.
However, recall that in general $J\neq I(V(J))$, as this requires the ideal to
be radical (see Eq. (29)). In fact, it turns out that the algebraic analogue
of a minimal decomposition of $U$, called a minimal primary decomposition,
expresses $J$ as

$J=\bigcap_{l=1}^{n_{Q}(J)}Q_{l}\,,$ (83)

where each $Q_{l}$ is primary (see Appendix A for the definition), all
$\sqrt{Q_{l}}$ are distinct, and no $Q_{l}$ can be removed from the
intersection without changing the result, i.e.$\;Q_{m}\nsupseteq\bigcap_{l\neq
m}Q_{l}$. We stress that the intersection of ideals should be viewed
considering the ideals as infinite sets of polynomials. We call each $Q_{l}$ a
primary component of $J$ and we denote the number of primary components as
$n_{Q}(J)$. The radical of each primary component $Q_{l}$ is a prime ideal

$P_{l}=\sqrt{Q_{l}}\,,$ (84)

i.e. a primary ideal is also prime only if it is radical. As $P_{l}$ is the
prime associated to $Q_{l}$, we say that $Q_{l}$ is $P_{l}\text{-primary}$. We
call the set of primes associated to all the primary components of $J$ the set
of associated primes. To this end, we write

$\mathrm{assoc}(J)=\big{\\{}P_{1},\ldots,P_{n_{Q}(J)}\big{\\}}\,.$ (85)

It can be shown that the associated primes in a minimal primary decomposition
are unique (books/daglib/0091700, , Theorem 4.5). We call an associated prime
$P_{i}$ of $J$ a minimal prime if $P_{i}\not\supseteq P_{j}$ for all $i\neq
j$. We define the set of minimal associated primes of $J$ as

$\mathrm{minAssoc}(J)=\Big{\\{}P\in\mathrm{assoc}(J)\text{ where }P\text{ is a
minimal prime}\Big{\\}}\,.$ (86)

It can be shown that the set of primary components $Q_{k}$ of $J$ for which
$\sqrt{Q_{k}}$ is a minimal prime of $J$ is unique (books/daglib/0091700, ,
Theorem 4.10).

Let us now address the relation between the minimal primary decomposition of
an ideal $J$ and the minimal decomposition of its associated variety $U=V(J)$.
First of all, note that the dimension of each primary component may not be the
same. In fact, it can be shown that

$\dim(J)=\max\left(\\{\dim(Q_{l})\,\,:\,\,1\leq l\leq{n_{Q}(J)}\\}\right)\,.$
(87)

If we now interpret Eq. (83) geometrically by taking the variety of both left-
and right-hand side and using the fact that the variety associated to an
intersection of a set of ideals corresponds to the union of the varieties
associated to each ideal, we obtain

$V(J)=\bigcup_{l=1}^{n_{Q}(J)}V(Q_{l})\,.$ (88)

In general, the union of Eq. (88) may not be a minimal decomposition of
$V(J)$, because the variety associated to a primary component may be contained
in the variety associated to another. Therefore, we can split set of varieties
$V(Q_{l})$ into two distinct subsets: those who can be removed from the
intersection of Eq. (88) without changing the result and those that cannot. We
refer to these as “embedded” and “isolated”, respectively. It can be shown
that the prime ideals $P_{k}$ in the set $\mathrm{minAssoc}(J)$ are in one-to-
one correspondence with the irreducible varieties $U_{k}$ from Eq. (81), i.e.
$U_{k}=V(P_{k})$. Therefore, it is clear that $n_{Q}(J)\geq n_{B}(V(J))$.
Geometrically, one can see that the non-uniqueness in a minimal primary
decomposition is associated to the primary ideals $Q_{l}$ such that $V(Q_{l})$
is embedded.

Before moving on to some explicit examples in spinor space, let us remark that
there exist general algorithms for the computation of primary decompositions,
see for instance Ref. gianni1988grobner . A further useful comment is that for
a prime ideal, all maximally independent sets are of the same size
(becker2012groebner, , Proposition 7.26). This observation can provide a
simple way to show that an ideal is not prime.

##### Examples of Irreducible Singular Varieties At Three and Four Points.

To understand Eqs. (83) and (88) in a more physical context, let us turn to
simple examples of the problem at hand: understanding surfaces in spinor
space. As a first warm-up, we consider there-point phase space. A well-known
fact is that either all angle or all square brackets must be zero. Formally,
this means that the zero ideal in $R_{3}$ (or equivalently the ideal in
$S_{3}$ generated by momentum conservation alone) is not primary. That is, one
can compute the primary decomposition of
$\bigl{\langle}0\bigr{\rangle}_{R_{3}}$ to find

$\displaystyle\bigl{\langle}0\bigr{\rangle}_{R_{3}}$ $\displaystyle=$
$\displaystyle\bigl{\langle}\langle 12\rangle,\langle 13\rangle,\langle
23\rangle\bigr{\rangle}_{R_{3}}\cap\bigl{\langle}[12],[13],[23]\bigr{\rangle}_{R_{3}}\,.$
(89)

In contrast, four-point phase space, $V\left(\langle 0\rangle_{R_{4}}\right)$,
is irreducible. However, a number of interesting varieties associated to
codimension-one ideals do decompose. This is again a statement that we can
demonstrate with the help of a primary decomposition. For instance, we have

$\bigl{\langle}\,[12]\,\bigr{\rangle}_{R_{4}}=\bigl{\langle}[12],\langle
34\rangle\bigr{\rangle}_{R_{4}}\cap\bigl{\langle}[12],[13],[14],[23],[24],[34]\bigr{\rangle}_{R_{4}}\,.$
(90)

One way to see why $\bigl{\langle}[12]\bigr{\rangle}_{R_{4}}$ must decompose
is to note that in $R_{4}$

$\langle 12\rangle[12]=\langle 34\rangle[34]\,,$ (91)

i.e. $\langle 34\rangle[34]$ is a member of
$\bigl{\langle}[12]\bigr{\rangle}_{R_{4}}$, but this is not the case for
$\langle 34\rangle$ nor $[34]$. Therefore, we must have at least two branches,
each one containing one of the two factors of $s_{34}$.

A well-known fact that can be interpreted in terms of this splitting is that
four-point massless amplitudes have non-unique common denominators in terms of
spinor brackets. To better understand this let us consider as a concrete
example the Parke-Taylor expression PhysRevLett.56.2459  for maximally-
helicity-violating (MHV) and $\overline{\text{MHV}}$ trees, say

$iA_{g^{-}g^{-}g^{+}g^{+}}=\frac{\,\langle 12\rangle^{3}}{\langle
23\rangle\langle 34\rangle\langle
41\rangle}=\frac{\,[34]^{3}}{[12][23][41]}\,.$ (92)

As shown in Eq. (92), at four points MHV and $\overline{\text{MHV}}$ coincide.
Thus, the denominator is clearly not unique. To see this algebro-
geometrically, let us begin by posing an apparently legitimate question; that
is, whether $\langle 23\rangle$ is a pole of this amplitude. We can now say
that this question is ill-posed because the surface is reducible. The
amplitude $A_{g^{-}g^{-}g^{+}g^{+}}$ has a simple pole on
$V\big{(}\bigl{\langle}\,\langle
23\rangle,[14]\,\bigr{\rangle}_{R_{4}}\big{)}$ but it is regular on
$V\big{(}\bigl{\langle}\langle 12\rangle,\langle 13\rangle,\langle
14\rangle,\langle 23\rangle,\langle 24\rangle,\langle
34\rangle\bigr{\rangle}_{R_{4}}\big{)}$; that is, it has a different behavior
on the different branches of $V\big{(}\bigl{\langle}\,\langle
23\rangle\,\bigr{\rangle}_{R_{4}}\big{)}$. Therefore, $\langle 23\rangle$ is
both a physical and a spurious singularity, depending on the branch we look
at. In conclusion, the physical statement about the singularity is that
$A_{g^{-}g^{-}g^{+}g^{+}}$ has a simple pole on
$V\big{(}\bigl{\langle}\,\langle
23\rangle,[14]\,\bigr{\rangle}_{R_{4}}\big{)}$ and algebraically we can
represent this in two different ways, either via $\langle 23\rangle$ or via
$[14]$ in the denominator.

As a final warm-up, let us consider the set of ideals at four points generated
by pairs of invariants, together with their primary decompositions. We present
a set of such ideals from which all others can be derived by permutations of
the $n$ momenta and parity.

$\bigl{\langle}\langle 12\rangle,\langle 13\rangle\bigr{\rangle}_{R_{4}}$ | $=$ | $P_{2}\cap P_{3}\cap\overline{P}_{3}(2341)\,,$ | $P_{1}$ | $=$ | $\bigl{\langle}\langle 12\rangle,[34]\bigr{\rangle}_{R_{4}}\,,$ |
---|---|---|---|---|---|---
$\bigl{\langle}\langle 12\rangle,\langle 34\rangle\bigr{\rangle}_{R_{4}}$ | $=$ | $P_{2}\cap P_{4}\,,$ | $P_{2}$ | $=$ | $\bigl{\langle}\langle 12\rangle,\langle 13\rangle,\langle 14\rangle,\langle 23\rangle,\langle 24\rangle,\langle 34\rangle\bigr{\rangle}_{R_{4}}\,,$ |
$\bigl{\langle}\langle 12\rangle,[12]\bigr{\rangle}_{R_{4}}$ | $=$ | $P_{4}\cap P_{5}\cap\overline{P}_{5}\,,$ | $P_{3}$ | $=$ | $\bigl{\langle}\langle 12\rangle,\langle 13\rangle,\langle 23\rangle,\tilde{\lambda}_{4}^{\dot{\alpha}}\bigr{\rangle}_{R_{4}}\,,$ |
$\bigl{\langle}\langle 12\rangle,[13]\bigr{\rangle}_{R_{4}}$ | $=$ | $P_{3}(1243)\cap\overline{P}_{3}(1342)\,$ | $P_{4}$ | $=$ | $\bigl{\langle}\langle 12\rangle,[12],\langle 34\rangle,[34]\,,$ | $(93)$
|  | $\quad\cap P_{5}\cap\overline{P}_{5}(1324)\,,$ |  |  | $\quad\lambda_{1}^{\alpha}\tilde{\lambda}_{1}^{\dot{\alpha}}+\lambda_{2}^{\alpha}\tilde{\lambda}_{2}^{\dot{\alpha}},\lambda_{3}^{\alpha}\tilde{\lambda}_{3}^{\dot{\alpha}}+\lambda_{4}^{\alpha}\tilde{\lambda}_{4}^{\dot{\alpha}}\bigr{\rangle}_{R_{4}}\,,$ |
$\bigl{\langle}\langle 12\rangle,[34]\bigr{\rangle}_{R_{4}}$ | $=$ | $P_{1}\,,$ | $P_{5}$ | = | $\bigl{\langle}\langle 12\rangle,[12],[13],[14],[23],[24],[34]\bigr{\rangle}_{R_{4}}\,.$ |

Here, on the left-hand side, we show ideals generated by pairs of invariants
together with their primary decompositions; on the right-hand side, we give
minimal bases for their primary components. We remark that $P_{3}$, $P_{4}$
and $P_{5}$ have codimension two, while $P_{1}$ and $P_{2}$ have codimension
one.

###  2.4 Functions Vanishing to Higher Order on Singular Varieties

Now that we have discussed how to construct the set of irreducible singular
varieties of rational prefactors, our aim is to use these varieties to study
the prefactors and interpret their behavior close to singular varieties as
constraints on the analytic structure of their numerators. In this section, we
review a well-studied class of ideals that we can use to parameterize these
constraints. Specifically, we introduce the so-called “symbolic power” of an
ideal (see Ref. dao2017symbolic  for a recent survey and Chapter 3.9 of Ref.
eisenbud1995commutative  for a textbook discussion). Importantly, we can use
the symbolic power to define what we mean by “behavior close to a variety”.
Our ultimate goal is to present a numerical algorithm for this study. However,
in this section we content ourselves with the mathematical preliminaries, and
delay the discussion of the numerical procedure to Section 3.

##### Vanishing to Higher Order at a Point.

Our goal is to discuss a set of elements of $R_{n}$ that vanish to
$k^{\mathrm{th}}$ order on a variety $U\subset V(J_{\Lambda_{n}})$. Before we
tackle this problem, we begin by studying the simpler case where the variety
is a single point. Let us consider a point $(\eta,\tilde{\eta})$ in spinor
space that satisfies momentum conservation, i.e. $(\eta,\tilde{\eta})\in
V(J_{\Lambda_{n}})$. The set of elements of $R_{n}$ that vanish on this point
is given by an ideal

$\mathfrak{m}_{(\eta,\tilde{\eta})}=\left\langle\lambda_{10}-\eta_{10},\lambda_{11}-\eta_{11},\ldots,\tilde{\lambda}_{1\dot{0}}-\tilde{\eta}_{1\dot{0}},\tilde{\lambda}_{1\dot{1}}-\tilde{\eta}_{1\dot{1}},\ldots\right\rangle_{R_{n}}.$
(94)

Here we label such an ideal as $\mathfrak{m}_{(\eta,\tilde{\eta})}$, hinting
that it is actually a so-called “maximal ideal” (see Appendix A for the
algebraic definition). To better understand Eq. (94), let us consider
computing the remainder of an element $q\in R_{n}$ modulo a Gröbner basis of
$\mathfrak{m}_{(\eta,\tilde{\eta})}$. From Eq. (94), it is easy to see that
the remainder modulo $\mathcal{G}(\mathfrak{m}_{(\eta,\tilde{\eta})})$ of $q$
is equivalent to the evaluation of $q$ at the point ${(\eta,\tilde{\eta})}$,
that is

$\Delta_{\mathcal{G}(\mathfrak{m}_{(\eta,\tilde{\eta})})}(q)=q{(\eta,\tilde{\eta})}\,.$
(95)

With this perspective, it is clear that $q$ vanishes at the point
${(\eta,\tilde{\eta})}$ if and only if it belongs to the ideal
$\mathfrak{m}_{(\eta,\tilde{\eta})}$.

With the ideal $\mathfrak{m}_{(\eta,\tilde{\eta})}$ in hand, we are now in a
position to define a set of elements of $R_{n}$ that vanish to
$k^{\mathrm{th}}$ order at the point ${(\eta,\tilde{\eta})}$. To motivate the
definition, let us start by noting that it is natural to say that elements of
$\mathfrak{m}_{(\eta,\tilde{\eta})}$ vanish to (at least) first order at the
point $(\eta,\tilde{\eta})$. One way to think of this is to consider a point

$(\eta^{(\epsilon)},\tilde{\eta}^{(\epsilon)})=(\eta+\epsilon\delta,\tilde{\eta}+\epsilon\tilde{\delta})\,,$
(96)

for some small quantity $\epsilon$ and a point $(\delta,\tilde{\delta})$ in
spinor space which is not required to satisfy momentum conservation itself,
but is chosen such that $(\eta^{(\epsilon)},\tilde{\eta}^{(\epsilon)})$
satisfies momentum conservation. At this shifted point
$(\eta^{(\epsilon)},\tilde{\eta}^{(\epsilon)})$, the generators of
$\mathfrak{m}_{(\eta,\tilde{\eta})}$ are all proportional to $\epsilon$ and we
can interpret this as vanishing to first order. It is easy to see that if we
raise $\mathfrak{m}_{(\eta,\tilde{\eta})}$ to $k^{\mathrm{th}}$ power then all
of the generators of this power ideal will be proportional to $\epsilon^{k}$.
This leads us to define that an element $q\in R_{n}$ vanishes to
$k^{\mathrm{th}}$ order at a point $(\eta,\tilde{\eta})$, if it is an element
of $\mathfrak{m}_{(\eta,\tilde{\eta})}^{k}$, i.e.

$q\in\mathfrak{m}_{(\eta,\tilde{\eta})}^{k}\;\;\Rightarrow\;\;q\text{ vanishes
to }k^{\mathrm{th}}\text{ order at }{(\eta,\tilde{\eta})}\,.$ (97)

We remind the reader that the ideal power is computed by repeated
multiplication of the generators, see Eqs. (21) and (22).

##### Vanishing to Higher Order on a Variety.

Let us now consider elements of $R_{n}$ that vanish not just at a single
point, but on an entire variety $W$. We ask an analogous question to the case
where the variety was a single point, that is whether we can construct the set
of elements that vanish to $k^{\mathrm{th}}$ order at every point on $W$.
Given our previous discussion, we therefore want to understand elements which
belong to ${\mathfrak{m}}_{(\eta,\tilde{\eta})}^{k}$ for every
$(\eta,\tilde{\eta})\in W$. This is the intersection of each of these ideals,
that is

$\bigcap_{(\eta,\tilde{\eta})\in
W}{\mathfrak{m}}_{(\eta,\tilde{\eta})}^{k}\,.$ (98)

We will refer to this intersection as the set of elements of $R_{n}$ which
vanish to $k^{\mathrm{th}}$ order on $W$. As the set of points in $W$ is
potentially infinite, the computation of this intersection is a non-trivial
exercise. A natural expectation is that the set of elements which vanish to
$k^{\mathrm{th}}$ order on $W$ is related to $I(W)^{k}$. However, it turns out
that $I(W)^{k}$ is insufficient: there can exist elements of $R_{n}$ which
vanish to $k^{\mathrm{th}}$ order but do not belong to $I(W)^{k}$. We must
introduce a refined definition of ideal power: the so-called symbolic power
(see Chapter 3.9 eisenbud1995commutative ).

Let us begin with an irreducible variety $U$. In the case of a prime ideal
such as $I(U)$, the symbolic power can be defined as the $I(U)\text{-primary}$
component of the ideal power. More precisely, we consider the minimal primary
decomposition of $I(U)^{k}$ which we can write as888It is perhaps surprising
that there could be multiple primary components $Q_{i}$, since
$\sqrt{I(U)^{k}}=I(U)$, but this is possible because in general
$V(Q_{i})\subseteq U$. That is, it is possible that $V(Q_{i})$ can be
embedded. This corresponds to the fact that the ideal $I(U)^{k}$ does not
necessarily contain all functions that vanish to $k^{\mathrm{th}}$ order on
$U$.

$I(U)^{k}=\bigcap_{i=1}^{m}Q_{i}\,.$ (99)

The $k^{\mathrm{th}}$ symbolic power of an ideal associated to an irreducible
variety $U$ is defined as

$I(U)^{\langle k\rangle}=Q_{j}\,,\;\,\text{where}\;\;\sqrt{Q_{j}}=I(U)\,,$
(100)

that is, the $k^{\text{th}}$ symbolic power of $I(U)$ is the unique primary
component $Q_{j}$ of $I(U)^{k}$ whose associated prime is $I(U)$. It is clear
from the definition that $I(U)^{\langle 1\rangle}=I(U)$. Consider now a
situation where we work with a reducible variety $W$. Then the symbolic power
can be defined as the intersection of the symbolic powers of the ideals
associated to the irreducible components of $W$, i.e.

$I(W)^{\langle
k\rangle}=\bigcap_{P_{i}\,\in\,\mathrm{assoc}(I(W))}P_{i}^{\langle
k\rangle}\,.$ (101)

We are now prepared to describe the set of elements of $R_{n}$ which vanish to
$k^{\mathrm{th}}$ order on a variety. The key theorem we need is the so-called
“Zariski–Nagata theorem” Zariski1949 ; Nagata1962 , in the general form
introduced by Eisenbud and Hochster EISENBUD1979157 . For our purposes, it
states that for a radical ideal $J$ in $R_{n}$

$\bigcap_{(\eta,\tilde{\eta})\in
V(J)}{\mathfrak{m}}_{(\eta,\tilde{\eta})}^{k}\subseteq J^{\langle
k\rangle}\,.$ (102)

where we stress that the powers of the maximal ideals, and the symbolic power
of $J$, are computed in $R_{n}$. We see that Zariski–Nagata tells us that the
set of elements of $R_{n}$ that vanish to $k^{\mathrm{th}}$ order on the
variety $V(J)$ is contained within the $k^{\mathrm{th}}$ symbolic power of
$J$. Therefore, we see that if we wish to compute the set of polynomials that
vanish on a variety $W$ to $k^{\mathrm{th}}$ order, it is sufficient to
compute $I(W)^{\langle k\rangle}$.

##### Computing Symbolic Powers.

A natural question is how one computes the symbolic power in practice. It is
clear from Eq. (100) that, for a prime ideal $P$, one can calculate the
$k^{\mathrm{th}}$ symbolic power $P^{\langle k\rangle}$ by computing the
primary decomposition of $P^{k}$. However, obtaining the primary decomposition
can be computationally demanding. In order to circumvent this, we now
introduce a useful lemma. First, we note an important technical property of
$R_{n}$. As it is a quotient of a polynomial ring by a maximal codimension
ideal, $R_{n}$ is “Cohen–Macaulay” (eisenbud1995commutative, , Proposition
18.13). This property has a very useful consequence for certain ideals when
computing symbolic powers. Specifically, if $A$ is a Cohen–Macaulay ring and
$J$ is a maximal codimension ideal of $A$ then the ideal power and symbolic
power coincide (zariski2013commutative, , Appendix 6, Lemma 5). That is, for
an ideal $J$ of $R_{n}$ we have

$\text{codim}(J)=\mu(J)\;\;\Rightarrow\;\;J^{\langle n\rangle}=J^{n}\,.$ (103)

One can understand this as follows: if the ideal associated to a variety $U$
is of maximal codimension, then the functions which vanish to
$k^{\mathrm{th}}$ order on $U$ are simply given by $I(U)^{k}$. In many cases,
strategic application of the lemma in Eq. (103) allows one to bypass the
computation of unnecessary primary decompositions when computing symbolic
powers.

##### Examples of Symbolic Powers.

To build intuition, let us reconsider the prime ideals $P_{1}$ through $P_{5}$
in the four-point quotient ring $R_{4}$, as given in Eq. (2.3). It can be
shown that

$\big{\langle}\langle 12\rangle,[34]\big{\rangle}_{R_{4}}^{\langle
2\rangle}=\big{\langle}\langle
12\rangle,[34]\big{\rangle}_{R_{4}}^{2}=\big{\langle}\langle
12\rangle^{2},\langle 12\rangle[34],[34]^{2}\big{\rangle}_{R_{4}}\,.$ (104)

That is, in this case, the second symbolic power agrees with the second ideal
power. In fact, for almost all of the $P_{i}$ in Eq. (2.3) this holds.
Specifically,

$\text{assoc}(P_{i}^{2})=\\{P_{i}\\}\quad\Longrightarrow\quad
P_{i}^{2}=P_{i}^{\langle 2\rangle}\quad\forall\;i\neq 5\,.$ (105)

That is, the second ideal power corresponds to the second symbolic one in all
cases except for $P_{5}$. Let us then consider the case of $P_{5}$, where the
symbolic power does not coincide with the normal power. The associated primes
are

$\text{assoc}(P_{5}^{2})=\\{P_{5},P_{\text{x}}\\}\quad\text{with}\quad
P_{\text{x}}=\big{\langle}\langle ij\rangle,[ij]\;\,\,:\,\,1\leq i<j\leq
n\big{\rangle}_{R_{4}}\,,$ (106)

and the primary decomposition reads

$P_{5}^{2}=Q_{5}\cap
Q_{\text{x}}\quad\text{with}\quad\sqrt{Q_{5}}=P_{5}\,,\;\sqrt{Q_{\text{x}}}=P_{\text{x}}\quad\Longrightarrow\quad
P_{5}^{\langle 2\rangle}=Q_{5}\,.$ (107)

One finds that the size of the minimal generating sets are given by
$\mu(Q_{5})=16$ and $\mu(Q_{x})=49$. We, therefore, do not print these ideals
in the text, but they are easily obtainable with computer algebra techniques.
We note that there must be some polynomial which belongs to the symbolic power
$P_{5}^{\langle 2\rangle}$ but not to $Q_{\text{x}}$, and hence not to
$P_{5}^{2}$. It is easy to check that

$[34]\notin Q_{\text{x}}\quad\text{and}\quad[34]\notin
P_{5}^{2}\,,\quad\text{but}\quad[34]\in P_{5}^{\langle 2\rangle}\,.$ (108)

##  3 Numerical Points near Singular Varieties

In the previous section, we introduced the class of polynomials which vanish
to a $k^{\mathrm{th}}$ order on a variety. We now wish to understand how to
generate numerical configurations of spinors that are close to irreducible
singular varieties. This will allow us to numerically determine the degree of
vanishing of a numerator polynomial. One of the important properties of finite
fields that makes them useful in computer algebra applications is that, in
contrast to real or complex numbers, they can be exactly represented on a
computer without approximation. However, if we wish to use finite fields to
construct configurations of spinors which are close to some other
configuration, this is not possible as it turns out that the available measure
of size is not sufficiently powerful. To understand this mathematically, we
now review the idea of an absolute value on a field $\mathbb{F}$ which will
allow us to formalize the notion of size. Absolute values on fields are a
basic idea in the theory of number fields and we refer to Ref. Gouvea1997  for
an introduction. Mathematically, when we wish to discuss the size of elements
of a field $\mathbb{F}$ we make use of a map $|\cdot|_{\mathbb{F}}$ from a
field $\mathbb{F}$ to the non-negative real numbers $\mathbb{R}_{\geq 0}$,
known as an absolute value999We note that a field $\mathbb{F}$ may admit
multiple absolute values, but for the cases in this work it will be clear by
context the one which we consider.. Well-known absolute values include the
standard ones on the real and complex numbers. If we have two elements $x$ and
$y$ of a field $\mathbb{F}$, we will say that $x$ is smaller than $y$ if

$|x|_{\mathbb{F}}\,<\,|y|_{\mathbb{F}}\,.$ (109)

Note that the result of $|\cdot|_{\mathbb{F}}$ is always a real number, so the
comparison in Eq. (109) takes place in the real numbers. An absolute value
also induces a metric $d$ on ${\mathbb{F}}$, given by

$d(x,y)=|x-y|_{\mathbb{F}}\,,$ (110)

where $x$ and $y$ are two elements of ${\mathbb{F}}$. We will mostly make use
of Eq. (109)—the ability to compare sizes of elements of a field—in order to
discuss points close to a variety.

Let us return to the finite-field case. It can be shown that the only absolute
value on $\mathbb{F}_{p}$ is the so-called trivial absolute value which takes
one of two values Gouvea1997 . That is, for all $a\in\mathbb{F}_{p}$ one can
show that101010This is easily proven by using Fermat’s little theorem and by
multiplicativity and non-negativity of the absolute value. Let
$a\in\mathbb{F}_{p}$, then:
$a^{p}=a\;\Rightarrow\;|a^{p}|-|a|=0\;\Rightarrow\;|a|^{p}-|a|=0\;\Rightarrow\;|a|(|a|^{p-1}-1)=0\;\Rightarrow\;|a|=0\;\lor\;|a|=1$.
Finally, by positive-definiteness we have $|a|=0\;\Rightarrow\;a=0$, and hence
Eq. (111).

$|a=0|_{\mathbb{F}_{p}}=0\quad\text{or}\quad|a\neq 0|_{\mathbb{F}_{p}}=1\,.$
(111)

Considering the induced metric on $\mathbb{F}_{p}$, one can then say that two
elements $x$ and $y$ of $\mathbb{F}_{p}$ are either $0$ or $1$ units apart.
This implies that in $\mathbb{F}_{p}$ we can only generate phase-space points
which are either on or away from a given surface. Therefore, the induced
metric does not admit a non-trivial hierarchy of distances.

In order to bypass this issue, in Section 3.1 we introduce another number-
theoretical field that admits a more powerful measure of distance: the $p\kern
0.80002pt\text{-adic}$ numbers. Then, in Sections 3.2 and 3.3, we show how to
start from a finite-field-valued configuration of spinors that is on a variety
to then construct a $p\kern 0.80002pt\text{-adic}$ configuration of spinors
which is close to said variety, by perturbing the finite-field configuration.

###  3.1 Beyond Finite Fields: $p\kern 0.80002pt\text{-adic}$ Numbers

In this section, we review mathematical details of the $p\kern
0.80002pt\text{-adic}$ numbers that are relevant for our applications. These
are well-studied objects in the mathematical literature and we refer the
reader to textbooks such as Ref. Gouvea1997  for a pedagogical introduction to
the topic. We begin by introducing the so-called $\boldsymbol{p}\kern
0.80002pt\text{-adic}$ integers, which we denote as $\mathbb{Z}_{p}$. These
are not to be confused with a finite field with $p$ elements, which we denote
as $\mathbb{F}_{p}$. An element $z\in\mathbb{Z}_{p}$ can be considered as a
power series in a prime number $p$, i.e.

$z=\sum_{i=0}^{\infty}a_{i}p^{i}=a_{0}+a_{1}p+a_{2}p^{2}+\cdots\,,$ (112)

where the $a_{i}$ take integer values in the range $[0,p-1]$. We call the
coefficients $a_{i}$ the $\boldsymbol{p}\kern 0.80002pt\text{-adic}$ digits of
$z$, in analogy to a decimal representation of a real number. Multiplication
and addition of elements of $\mathbb{Z}_{p}$ can be defined using the standard
multiplication and addition rules for power series. However, one must also
take into account that the digits of the resulting series must still live in
the range $[0,p-1]$. This can always be achieved by carry rules, analogous to
performing arithmetic with decimal numbers. With this in mind, the first non-
zero $p\kern 0.80002pt\text{-adic}$ digit behaves like an element of
$\mathbb{F}_{p}$. It can be shown that the set of $p\kern
0.80002pt\text{-adic}$ integers forms a ring under multiplication and
addition. However, $\mathbb{Z}_{p}$ is not a field: there exists no
multiplicative inverse for any element of $\mathbb{Z}_{p}$ with zero as its
first $p\kern 0.80002pt\text{-adic}$ digit.

Let us reconsider the power series representation of a $p\kern
0.80002pt\text{-adic}$ integer given in Eq. (112). Note that extending this
representation to allow for negative powers of $p$ solves the issue that
prevents the $p\kern 0.80002pt\text{-adic}$ integers from being a field. This
leads us to the $\boldsymbol{p}\kern 0.80002pt\text{-adic}$ numbers, which we
denote as $\mathbb{Q}_{p}$. Specifically, an element $x\in\mathbb{Q}_{p}$
takes the form

$x=\sum_{i=-l}^{\infty}a_{i}p^{i}=a_{-l}p^{-l}+\cdots+a_{-1}p^{-1}+a_{0}+a_{1}p+a_{2}p^{2}+\cdots\,,$
(113)

where again $a_{i}$ is an integer in the range $[0,p-1]$. Multiplication and
addition are again defined by power series operations with carries. It is
important to note that the $p\kern 0.80002pt\text{-adic}$ numbers are not an
algebraically closed field.

As promised, the $p\kern 0.80002pt\text{-adic}$ numbers are a field with a
more powerful way to measure size. To discuss this, given a $p\kern
0.80002pt\text{-adic}$ number $x$, we first introduce the $\boldsymbol{p}\kern
0.80002pt\text{-adic}$ valuation of $x$, which we denote by $\nu_{p}(x)$.
Considering the power series representation of $x\in\mathbb{Q}_{p}$ of Eq.
(113), the valuation of a non-zero $x$ is the integer $k$ such that $a_{k}$ is
the first non-zero $p\kern 0.80002pt\text{-adic}$ digit of $x$. That is,

$\nu_{p}(x)=k\,\text{ such that }\,a_{i}=0\,\text{ for all }\,i<k\,.$ (114)

For $x=0$ it is conventional to take $\nu_{p}(x)=\infty$. The
$\boldsymbol{p}\kern 0.80002pt\text{-adic}$ absolute value, which we denote by
$|x|_{p}$, is defined as

$|x|_{p}=p^{-\nu_{p}(x)}\,,$ (115)

for $x\neq 0$, and $|0|_{p}=0$. It is this absolute value on $\mathbb{Q}_{p}$
that we will use to discuss size.

This measurement of size has a number of interesting implications. Firstly, we
see that $p\kern 0.80002pt\text{-adic}$ numbers which are proportional to $p$
are $p\kern 0.80002pt\text{-adically}$ small, and those proportional to
$\frac{1}{p}$ are $p\kern 0.80002pt\text{-adically}$ large. That is,
considering $p$ as a $p\kern 0.80002pt\text{-adic}$ number, we have

$|p|_{p}<1<\left|\frac{1}{p}\right|_{p}\,.$ (116)

We emphasize that, when considered $p\kern 0.80002pt\text{-adically}$, the
quantity $p$ is to be regarded as small. Note that the $p\kern
0.80002pt\text{-adic}$ integers form a subset of the $p\kern
0.80002pt\text{-adic}$ numbers whose absolute value is bounded from above.
That is,

$|x|_{p}\leq 1\quad\text{for all}\quad x\in\mathbb{Z}_{p}\,.$ (117)

Next, note that the $p\kern 0.80002pt\text{-adic}$ absolute value is discrete
and unbounded when considered over the set of $p\kern 0.80002pt\text{-adic}$
numbers. This is in contrast to the trivial absolute value, which is discrete
but bounded to either 0 or 1; or the standard absolute value over $\mathbb{R}$
which is unbounded but continuous. Finally, we note that, while numbers in
$\mathbb{R}$ satisfy the triangle inequality

$|x+y|_{\mathbb{R}}\leq|x|_{\mathbb{R}}+|y|_{\mathbb{R}}\quad\text{for}\quad
x,y\in\mathbb{R}\,,$ (118)

those in $\mathbb{Q}_{p}$ satisfy the strong triangle inequality

$|x+y|_{p}\leq\text{max}(|x|_{p},|y|_{p})\quad\text{for}\quad
x,y\in\mathbb{Q}_{p}\,.$ (119)

Eq. (119) states that when one sums two $p\kern 0.80002pt\text{-adic}$
numbers, the result cannot be larger than either of the two summands. In
practice, this can be helpful for establishing bounds on the size of
intermediate stages of calculations, which can be important for numerical
stability. Furthermore, it is important to note that, for large $p$, the bound
in Eq. (119) is frequently saturated in practice. This can be seen by analogy
to finite-field computations. Specifically, in $\mathbb{F}_{p}$ it is well-
understood that a quantity accidentally evaluating to zero can be made less
probable by raising the value of $p$. As the first digit of a $p\kern
0.80002pt\text{-adic}$ number behaves like an element of a finite-field, we
see that this implies that, by working with large $p$, one can make it
improbable that such a quantity becomes accidentally small. As quantities
accidentally becoming small is an important source of precision loss in many
algorithms, this has important practical implications for numerical stability.

#####  $\boldsymbol{p}\kern 0.80002pt\text{-adics}$ on a Computer.

Since computers have finite memories, one can consider truncating the power
series expansion of a $p\kern 0.80002pt\text{-adic}$ number. Recalling the
form of a $p\kern 0.80002pt\text{-adic}$ number $x$ from Eq. (113), one can
truncate the series and write

$x=a_{-l}p^{-l}+\ldots+a_{-1}p^{-1}+a_{0}+a_{1}p+\cdots+a_{m-1}p^{m-1}+\mathcal{O}(p^{m})\,.$
(120)

This can be understood as a $p\kern 0.80002pt\text{-adic}$ analogue of real
numbers being represented by floating-point numbers of finite precision.
Comparing the truncated power series in Eq. (120) to the full series in Eq.
(113), we see that the error $\mathcal{O}(p^{m})$ made by truncating the power
series can be made small in a $p\kern 0.80002pt\text{-adic}$ sense: increasing
$m$ decreases the error as $|p^{m}|_{p}<|p^{m-1}|_{p}$. To make use of this on
a computer, we use a floating-point representation111111A public
implementation of $\mathbb{Q}_{p}$ can be found in Sage sagemath  or FLINT
flint .. Specifically, we write a truncated $p\text{-adic}$ number $x$ as

$x=p^{\nu_{p}(x)}\left(\sum_{i=0}^{k-1}a_{i}p^{i}+\mathcal{O}(p^{k})\right)\;\,\text{with}\;\,a_{i}\neq
0\,,$ (121)

where we call the prefactor $p^{\nu_{p}(x)}$ the exponent, the summation part
the mantissa and $k\in\mathbb{Z}_{>0}$ the working precision. The mantissa can
be stored as a positive integer modulo $p^{k}$ and so practical floating-point
$p\kern 0.80002pt\text{-adic}$ arithmetic is very similar to working modulo
$p^{k}$.

Let us consider basic arithmetic operations in the floating-point
representation. First, consider multiplying $x$ by another $p\kern
0.80002pt\text{-adic}$ number $y$, whose digits we denote as $b_{i}$. This is
given by

$xy=p^{\nu_{p}(x)+\nu_{p}(y)}\left[\left(\sum_{i=0}^{k-1}a_{i}p^{i}\right)\left(\sum_{i=0}^{k-1}b_{i}p^{i}\right)+\mathcal{O}(p^{k})\right]\,.$
(122)

Here, we can clearly identify the exponent of the product as the sum of the
exponents, and the mantissa of the product as the product of the mantissae
modulo $p^{k}$. The multiplicative inverse of $x$ can computed as

$x^{-1}=p^{-\nu_{p}(x)}\left(\overline{x}+\mathcal{O}(p^{k})\right)\,,$ (123)

where $\overline{x}$ is an integer satisfying

$np^{k}+\overline{x}\,\,\sum_{i=0}^{k-1}a_{i}p^{i}=1\,,$ (124)

for some auxiliary integer $n$. Such a pair $(\overline{x},n)$ can easily be
computed through the extended Euclidean algorithm applied to the mantissa and
$p^{k}$, in analogy to the finite-field case (see, for example, Ref.
(Peraro:2016wsq, , Appendix A)). Note that, as $p$ is prime, both
multiplication and the computation of multiplicative inverse have the property
that the mantissa of the result cannot be proportional to $p$, as required in
the floating-point representation of Eq. (121).

Let us now consider addition in the floating-point representation. In contrast
to multiplication, questions of stability arise. Without loss of generality we
can consider $\nu_{p}(y)\geq\nu_{p}(x)$ and write the summation of $x$ and $y$
as

$x+y=p^{\nu_{p}(x)}\left[\left(\sum_{i=0}^{k-1}a_{i}p^{i}\right)+p^{[\nu_{p}(y)-\nu_{p}(x)]}\left(\sum_{i=0}^{k-1}b_{i}p^{i}\right)+\mathcal{O}(p^{k})\right]\,.$
(125)

In comparison to multiplication, it is more subtle to compute the exponent and
mantissa of the sum from this form. Specifically, for the case where
$\nu_{p}(x)=\nu_{p}(y)$, the part in square brackets in Eq. (125) may be
proportional to $p$. This violates the assumption in Eq. (121) that the
leading digit of the mantissa is non-zero. To return to the floating-point
representation, one must then shuffle factors of $p$ from the mantissa to the
exponent. However, as the mantissa is only known to $k$ digits, this procedure
introduces an arbitrary choice into the last digits of the new mantissa. In
more traditional terms, one may lose precision when performing addition. In
practice, similarly to working in $\mathbb{F}_{p}$, this can be made unlikely
to accidentally happen by increasing the size of the prime $p$. We note that
this is a generalization of the issue of accidental division by zero in finite
fields.

###  3.2 Finite-Field Points on Singular Varieties

Let us now discuss how one can generate a point on a variety when working in
$\mathbb{F}_{p}$. To ease the discussion, we will work over the polynomial
ring $\mathbb{F}_{p}[X_{1},\ldots,X_{n}]$, and denote the tuple of variables
as $\underline{X}=\\{X_{1},\ldots,X_{n}\\}$. Given an ideal $J=\langle
q_{1},\ldots,q_{m}\rangle_{\mathbb{F}_{p}[\underline{X}]}$, we wish to
generate a numerical point $\underline{X}^{(0)}\in\mathbb{F}_{p}^{n}$ that is
a solution to the equations

$q_{i}(\underline{X})=0\quad\text{for}\quad i=1,\ldots,m\,.$ (126)

That is, $\underline{X}^{(0)}\in V(J)$. Clearly, for a variety that is not
zero dimensional, there are many such points $\underline{X}^{(0)}$. In the
following we will focus on constructing a single point $\underline{X}^{(0)}\in
V(J)$. Geometrically, our strategy is to intersect the variety with a randomly
chosen collection of hyperplanes so that this intersection is a zero-
dimensional variety. The zero-dimensional variety then corresponds to a finite
collection of points. We explicitly construct one such point and take this to
be $\underline{X}^{(0)}$.

In order to build the set of hyperplanes, we begin by constructing a maximally
independent set of $J$, as discussed in Section 2.1.2. We denote the maximally
independent set as $\underline{Y}$ and the corresponding dependent variables
as $\underline{Z}=\underline{X}\,\backslash\,\underline{Y}$. We remind the
reader that $\underline{Y}$ is a tuple of $\dim(J)$ variables and that
$\underline{Z}$ is a tuple of $\mathrm{codim}(J)$ variables. By definition,
the elements of the set $\underline{Y}$ can be chosen independently. If we
choose values for $\underline{Y}$ generically, then they specify a variety
such that its intersection with $V(J)$ is a zero-dimensional sub-variety of
$V(J)$. To this end, we construct a point
$\underline{Y}^{(0)}\in\mathbb{F}_{p}^{\dim(J)}$ by choosing each component
uniformly as integers from the range $[0,p-1]$. With this point in hand, we
now consider the system

$q_{i}(\underline{Z},\underline{Y}^{(0)})=0\,\quad\text{for}\quad
i=1,\ldots,m\,.$ (127)

This system of equations defines our zero-dimensional subvariety of $V(J)$.
Note that the polynomials in Eq. (127) $q_{i}$ depend only on the
$\underline{Z}$ variables, as the components of $\underline{Y}^{(0)}$ take
values in $\mathbb{F}_{p}$. It is useful to introduce the corresponding ideal
in the polynomial ring $\mathbb{F}_{p}[\underline{Z}]$, as

$J^{(0)}=\langle
q_{1}(\underline{Z},\underline{Y}^{(0)}),\ldots,q_{m}(\underline{Z},\underline{Y}^{(0)})\rangle_{\mathbb{F}_{p}[\underline{Z}]}\,.$
(128)

Clearly, any $\underline{Z}^{(0)}\in V(J^{(0)})$ can be combined with
$\underline{Y}^{(0)}$ to find our desired point $\underline{X}^{(0)}$.

Our problem is now reduced to the simpler task of finding an element of
$V(J^{(0)})$. However, in general the system of Eqs. (127) is non-linear in
$\underline{Z}$, which makes this a non-trivial exercise. To this end, we make
use of standard tools of elimination theory, which we now review. We refer the
reader to Chapter 3 of Ref. cox1994ideals  for a pedagogical introduction. The
key tool we will use is a Gröbner basis with a special monomial ordering. The
ordering that we need is the so-called lexicographic ordering on the variables
$\underline{Z}$, which we denote as $\succeq_{\text{lex.}}$. Specifically, we
order the variables as

$\succeq_{\text{lex.}}:Z_{\mathrm{codim}(J)}\succ\dots\succ Z_{1}\,.$ (129)

To highlight the use of this monomial order, we will denote the lexicographic
Gröbner basis of $J^{(0)}$ as $\mathcal{G}_{\text{lex.}}(J^{(0)})$. We now
consider the subset of $\mathcal{G}_{\text{lex.}}(J^{(0)})$ which depends only
on the variables $Z_{1}$ through $Z_{l}$. That is, we define

$\mathcal{G}_{l}=\mathcal{G}_{\text{lex.}}(J^{(0)})\cap\mathbb{F}_{p}[Z_{1},\ldots,Z_{l}]\,.$
(130)

The sets of polynomials $\mathcal{G}_{l}$ allow one to find a zero of
$\mathcal{G}_{\text{lex.}}(J^{(0)})$ in an iterative manner, constructing it
variable by variable. We will call a zero
$\\{Z_{1}^{(0)},\ldots,Z_{l}^{(0)}\\}$ of the polynomials $\mathcal{G}_{l}$ an
$l^{\mathrm{th}}$ partial solution. Note that one can always construct a
$0^{\mathrm{th}}$ partial solution as this is the empty set. Given a
$(l-1)^{\mathrm{th}}$ partial solution
$\\{Z_{1}^{(0)},\ldots,Z_{l-1}^{(0)}\\}$, our task is to find a
$Z_{l}^{(0)}\in\mathbb{F}_{p}$ such that
$\\{Z_{1}^{(0)},\ldots,Z_{l}^{(0)}\\}$ is an $l^{\mathrm{th}}$ partial
solution. We will refer to this as extending the $(l-1)^{\mathrm{th}}$ partial
solution. Clearly, repeatedly extending a partial solution will lead to an
element of $V(J^{(0)})$.

To discuss how to extend a partial solution, let us consider the ideal
generated by the evaluations of $\mathcal{G}_{l}$ on an $(l-1)^{\mathrm{th}}$
partial solution. That is, we consider the ideal

$J^{(0)}_{l,\text{eval}}=\left\langle
g(Z_{1}^{(0)},\ldots,Z_{l-1}^{(0)},Z_{l})\,\,:\,\,g\in\mathcal{G}_{l}\right\rangle_{\mathbb{F}_{p}[Z_{l}]}.$
(131)

It can be shown that $J^{(0)}_{l,\text{eval}}$ is generated by a single
polynomial $g_{o}$ (see Chapter 3.5 of Ref. cox1994ideals ). That is, one can
write $J^{(0)}_{l,\text{eval}}$ as

$J^{(0)}_{l,\text{eval}}=\langle g_{o}\rangle_{\mathbb{F}_{p}[Z_{l}]}\,.$
(132)

Note that a zero of $g_{o}$ is a $Z_{l}^{(0)}$ that allows us to extend the
$(l-1)^{\mathrm{th}}$ partial solution. Importantly, $g_{o}$ can be read from
$\mathcal{G}_{l}$. Specifically, let us write each element $g$ of
$\mathcal{G}_{l}$ in the form

$g=c_{g}(Z_{1},\ldots,Z_{l-1})Z_{l}^{N_{g}}+\text{ terms in which }Z_{l}\text{
has degree }<N_{g}\,.$ (133)

If we consider the set of polynomials $g^{\prime}\in\mathcal{G}_{l}$ such that
$c_{g^{\prime}}$ does not evaluate to zero on the $(l-1)^{\mathrm{th}}$
partial solution, then $g_{o}$ can be taken to be a $g^{\prime}$ which is a
non-constant polynomial and has lowest degree in $Z_{l}$ amongst all such
$g^{\prime}$. To extend a partial solution, we must therefore find a zero of
the univariate polynomial $g_{o}$. This can be solved systematically over
$\mathbb{F}_{p}$ by general, efficient algorithms such as the
Cantor–Zassenhaus algorithm cantor1981new . Note that, in principle, there may
be multiple zeros. As we only want a single point on the variety, it is
sufficient to take a single such zero.

In summary, starting from the trivial $0^{\mathrm{th}}$ partial solution, we
repeatedly extend the partial solution until we have constructed the
$\mathrm{codim}(J)^{\mathrm{th}}$ partial solution, which is the desired
$\underline{Z}^{(0)}$ which satisfies Eq. (127). This is then combined with
$\underline{Y}^{(0)}$, to give the desired $\underline{X}^{(0)}$. In general,
this procedure of extending a partial solution is only guaranteed to succeed
when working in an algebraically closed field, as this means $g_{o}$ must have
a zero in the field. This is relevant to our case as we work over
$\mathbb{F}_{p}$ which is not algebraically closed. We find a practical
solution to this problem is to repeat the procedure with different choices of
the $\underline{Y}^{(0)}$.

We make some final remarks. Firstly, we consider applying this procedure in
the case where $V(J)$ is reducible. This procedure will still generate a point
belonging to $V(J)$, however it provides no guarantee as to which branch of
$V(J)$ the point belongs. In practice we solve this issue by only applying the
approach to prime ideals. Secondly, there exist other algorithms to enumerate
the elements of $V(J^{(0)})$ that avoid the use of the lexicographic monomial
ordering: see, for example, Section 2.4 of Ref. cox2006using . This can prove
more efficient. However, we do not find this to be necessary in this work.

###  3.3 $p\kern 0.80002pt\text{-adic}$ Points Close to Singular Varieties

As already mentioned, we aim to evaluate rational functions on points in
spinor space which are $p\kern 0.80002pt\text{-adically}$ close to singular
varieties. This will allow us to numerically probe rational functions to learn
how fast they diverge or vanish. It this section we discuss how to obtain a
$p\kern 0.80002pt\text{-adic}$ point close to a given variety by perturbing an
exact finite-field solution. Thereafter, we discuss how one can interpret this
behavior in the language of algebraic geometry.

##### Lifting $\mathbb{F}\kern-6.5278pt\kern
0.18pt\mathbb{F}\kern-6.5278pt\kern 0.18pt\mathbb{F}_{\boldsymbol{p}}$
Solutions to the $\boldsymbol{p}\kern 0.80002pt\text{-adic}$ Integers.

Consider an ideal $J$ of $S_{n}$ that takes the form

$J=\langle q_{1},\ldots,q_{m},r_{1},\ldots,r_{4}\rangle_{S_{n}}\,,$ (134)

where $\\{r_{1},\ldots,r_{4}\\}$ generate $J_{\Lambda_{n}}$. Naturally,
$V(J)\subset V(J_{\Lambda_{n}})$ and we further assume that $J$ is prime. We
wish to construct a point
$(\eta^{(\epsilon)},\tilde{\eta}^{(\epsilon)})\in\mathbb{Z}_{p}^{4n}$, such
that

$\displaystyle\begin{split}q_{i}(\eta^{(\epsilon)},\tilde{\eta}^{(\epsilon)})&=\mathcal{O}(p)\quad\text{for}\quad
i=\\{1,\ldots,m\\}\,,\\\
r_{j}(\eta^{(\epsilon)},\tilde{\eta}^{(\epsilon)})&=\mathcal{O}(p^{k})\quad\text{for}\quad
j=\\{1,\ldots,4\\}\,,\end{split}$ (135)

where $k$ is a positive integer. We stress that a $p\kern
0.80002pt\text{-adic}$ integer point will be suitable for our purposes. As the
evaluations of the generators $q_{i}$ are $p\kern 0.80002pt\text{-adically}$
small, we consider such a point to be close to $V(J)$. As discussed in Section
3.1, when working with a computer we work with truncated $p\kern
0.80002pt\text{-adic}$ numbers. Therefore a solution to Eq. (135) is a point
that is close to $V(J)$, but on $V(J_{\Lambda_{n}})$ when working to $k$
digits of precision.

To construct our desired point, we will work digit by digit in the $p\kern
0.80002pt\text{-adic}$ expansion. Specifically, the $p\kern
0.80002pt\text{-adic}$ point in spinor space reads

$(\eta^{(\epsilon)},\tilde{\eta}^{(\epsilon)})=\left(\eta^{(\epsilon),0}+p\,\eta^{(\epsilon),1}+\ldots+\mathcal{O}(p^{k}),\tilde{\eta}^{(\epsilon),0}+p\,\tilde{\eta}^{(\epsilon),1}+\ldots+\mathcal{O}(p^{k})\right),$
(136)

where each of the $(\eta^{(\epsilon),i},\tilde{\eta}^{(\epsilon),i})$ are
integers in the range $[0,p-1]$. We will determine the
$(\eta^{(\epsilon),i},\tilde{\eta}^{(\epsilon),i})$ starting from $i=0$ and
moving up to the working precision.

The starting observation is that for
$(\eta^{(\epsilon)},\tilde{\eta}^{(\epsilon)})$ to be near $V(J)$, it must be
on $V(J)$ when truncated to first digit. We, therefore, begin with a finite-
field-valued configuration which lives on the variety analogous to $V(J)$ over
the finite fields, that is an
$(\eta,\tilde{\eta})_{\mathbb{F}_{p}}\in\mathbb{F}_{p}^{4n}$. Clearly, we can
use the algorithm of Section 3.2 to generate such a configuration.
Importantly, $(\eta,\tilde{\eta})_{\mathbb{F}_{p}}$ is a zero of the
generators of $J$ when considered modulo $p$. Therefore, we can reinterpret
each finite-field value as the first digit of a $p\kern 0.80002pt\text{-adic}$
integer. That is, we choose

$(\eta^{(\epsilon),0},\tilde{\eta}^{(\epsilon),0})=(\eta,\tilde{\eta})_{\mathbb{F}_{p}}\,,$
(137)

where we consider the components of $(\eta,\tilde{\eta})_{\mathbb{F}_{p}}$ to
be integers in the range $[0,p-1]$. We now have a $p\kern
0.80002pt\text{-adic}$ configuration which is a zero of the $q_{i}$ and
$r_{i}$ up to $\mathcal{O}(p)$ corrections. That is,

$q_{i}(\eta^{(\epsilon),0},\tilde{\eta}^{(\epsilon),0})=r_{i}(\eta^{(\epsilon),0},\tilde{\eta}^{(\epsilon),0})=\mathcal{O}(p)\,.$
(138)

Note that Eq. (138) implies that whatever the value of the digits
$(\eta^{(\epsilon),i},\tilde{\eta}^{(\epsilon),i})$ for $i>0$, the $q_{i}$
conditions of Eq. (135) will be satisfied. Therefore, we will not need to
consider the polynomials $q_{i}$ further. However, the present $p\kern
0.80002pt\text{-adic}$ point in spinor space does not yet satisfy momentum
conservation to working precision.

Our task, therefore, is to choose the remaining digits of
$(\eta^{(\epsilon)},\tilde{\eta}^{(\epsilon)})$ such that momentum
conservation is satisfied to $k$ digits. To achieve this, we work iteratively
order by order in $p$. For convenience, let us define

$(\eta^{(\epsilon),\overline{\nu}},\tilde{\eta}^{(\epsilon),\overline{\nu}})=\left(\sum_{i=0}^{\nu}p^{i}\eta^{(\epsilon),i},\sum_{i=0}^{\nu}p^{i}\tilde{\eta}^{(\epsilon),i}\right)\,.$
(139)

This represents the first $\nu+1$ digits of
$(\eta^{(\epsilon)},\tilde{\eta}^{(\epsilon)})$. Let us assume that we have
determined the $p\kern 0.80002pt\text{-adic}$ spinors
$(\eta^{(\epsilon),i},\tilde{\eta}^{(\epsilon),i})$ up to i = $\nu$. That is,
we assume that we have already fixed $\nu+1$ digits such that

$r_{i}(\eta^{(\epsilon),\overline{\nu}},\tilde{\eta}^{(\epsilon),\overline{\nu}})=\mathcal{O}(p^{\nu+1})\,.$
(140)

Our aim is to find a value for the next digit,
$(\eta^{(\epsilon),\nu+1},\tilde{\eta}^{(\epsilon),\nu+1})$ such that each
$r_{i}$ will vanish to one order higher in $p$. It turns out that
$(\eta^{(\epsilon),\nu+1},\tilde{\eta}^{(\epsilon),\nu+1})$ satisfies a system
of linear equations in a finite field. Let us expand the four $r_{i}$
polynomials around
$(\eta^{(\epsilon),\overline{\nu}},\tilde{\eta}^{(\epsilon),\overline{\nu}})$
to $\mathcal{O}(p^{\nu+2})$. One finds

$r_{i}(\eta^{(\epsilon)},\tilde{\eta}^{(\epsilon)})=r_{i}(\eta^{(\epsilon),\overline{\nu}},\tilde{\eta}^{(\epsilon),\overline{\nu}})+p^{\nu+1}\,\left[(\eta^{(\epsilon),\nu+1},\tilde{\eta}^{(\epsilon),\nu+1})\cdot\underline{\nabla}\,r_{i}\big{|}_{(\eta^{(\epsilon),\overline{\nu}},\tilde{\eta}^{(\epsilon),\overline{\nu}})}\right]+\mathcal{O}(p^{\nu+2})\,,$
(141)

where $\underline{\nabla}$ is the vector of derivatives with respect to
$(\lambda,\tilde{\lambda})$. If we now require that
$(\eta^{(\epsilon)},\tilde{\eta}^{(\epsilon)})$ is a zero of $r_{i}$ up to
$\mathcal{O}(p^{\nu+2})$ then we have a linear equation for the next digit.
That is,

$(\eta^{(\epsilon),\nu+1},\tilde{\eta}^{(\epsilon),\nu+1})\cdot\underline{\nabla}\,r_{i}\big{|}_{(\eta^{(\epsilon),\overline{\nu}},\tilde{\eta}^{(\epsilon),\overline{\nu}})}=-\frac{1}{p^{\nu+1}}r_{i}(\eta^{(\epsilon),\overline{\nu}},\tilde{\eta}^{(\epsilon),\overline{\nu}})+\mathcal{O}(p)\,.$
(142)

This is a linear system of equations for the next digit
$(\eta^{(\epsilon),\nu+1},\tilde{\eta}^{(\epsilon),\nu+1})$. At a practical
level, note that the constraints are modulo $p$, so they effectively give a
linear system in $\mathbb{F}_{p}$.

Importantly, the constraints in Eq. (142) always have a solution, given an
appropriate choice of $(\eta^{(\epsilon),0},\tilde{\eta}^{(\epsilon),0})$. We
can see this as follows. Note that the derivatives of the $r_{i}$ are being
evaluated close to the variety $V(J_{\Lambda_{n}})$ and so up to $O(p)$
corrections we can replace them with their evaluations on the variety. That
is,

$\underline{\nabla}\,r_{i}\big{|}_{(\eta^{(\epsilon),\overline{\nu}},\tilde{\eta}^{(\epsilon),\overline{\nu}})}=\underline{\nabla}\,r_{i}\big{|}_{(\eta,\tilde{\eta})}+O(p)\,,$
(143)

where $(\eta,\tilde{\eta})$ is a point on $V(J_{\Lambda_{n}})$. Therefore, up
to $O(p)$ corrections, the derivative vectors in Eq. (142) span the cotangent
space of $V(J_{\Lambda_{n}})$ at $(\eta,\tilde{\eta})$. As $J_{\Lambda_{n}}$
is a maximal codimension ideal this implies that, if $(\eta,\tilde{\eta})$ is
not a singular point of $V(J_{\Lambda_{n}})$, the linear system of equations
in Eq. (142) is of full rank and a solution exists. In practice it is easy to
avoid such singular points. Nevertheless, the system does not uniquely define
the value of $(\eta^{(\epsilon),\nu+1},\tilde{\eta}^{(\epsilon),\nu+1})$ as it
can be freely changed by any element of the tangent space of the
$V(J_{\Lambda_{n}})$ at the point $(\eta,\tilde{\eta})$. We make use of this
freedom and pick a random solution to Eq. (142).

Having determined the value of
$(\eta^{(\epsilon),\nu+1},\tilde{\eta}^{(\epsilon),\nu+1})$, we are now in a
position where we have a solution of Eq. (140) but with $\nu$ replaced with
$\nu+1$. It is therefore clear that we can iterate this procedure until we
find a solution with $\nu=k-1$, which is thus a solution to Eq. (135).

We close with a few remarks. Firstly, we point out that an analogous procedure
could be followed to generate points close to singular varieties when working
over $\mathbb{R}$ or $\mathbb{C}$. Secondly, let us also remark the similarity
of this multivariate procedure to the univariate Hensel’s lifting lemma.
Thirdly, it would be interesting to consider extending this procedure to
generate points in “asymmetric” approaches to a variety as employed in Ref.
DeLaurentis:2019phz .

##### Interpretation of $\boldsymbol{p}\kern 0.80002pt\text{-adic}$
Evaluations.

Let us now consider how to interpret the evaluation of a numerator
$\mathcal{N}\in R_{n}$ at a point close to an irreducible singular variety
$U$. Let $k$ be the largest integer such that
$\mathcal{N}\in\mathfrak{m}^{k}_{(\eta,\tilde{\eta})}$ holds for all points
$(\eta,\tilde{\eta})\in U$. For specific points $(\eta,\tilde{\eta})$, it may
be the case that $\mathcal{N}$ belongs to
$\mathfrak{m}^{k+1}_{(\eta,\tilde{\eta})}$, but these must always live on
higher codimension sub-varieties. We define

$\kappa(\mathcal{N},U)=k\,\,\text{ s.t.
}\,\left(\mathcal{N}\in\mathfrak{m}^{k}_{(\eta,\tilde{\eta})}\;\forall\;(\eta,\tilde{\eta})\in
U\right)\;\,\text{and}\;\,\left(\exists\;(\eta,\tilde{\eta})\in
U\,:\,\mathcal{N}\not\in\mathfrak{m}^{k+1}_{(\eta,\tilde{\eta})}\right)\,.$
(144)

Note that $\kappa(\mathcal{N},U)\geq 0$, as
$\mathcal{N}\in\mathfrak{m}^{0}_{(\eta,\tilde{\eta})}$ holds trivially.
Importantly, it is clear from the definition of $\kappa(\mathcal{N},U)$ that
we have

$\mathcal{N}\in\bigcap_{(\eta,\tilde{\eta})\in
U}\mathfrak{m}^{\kappa(\mathcal{N},U)}_{(\eta,\tilde{\eta})}\,,$ (145)

and we cannot replace $\kappa(\mathcal{N},U)$ with any higher integer. By the
Zariski–Nagata theorem we conclude that

$\mathcal{N}\in I(U)^{\langle{\kappa(\mathcal{N},U)}\rangle}\,.$ (146)

We will now argue that, for large $p$, a $p\kern 0.80002pt\text{-adic}$
evaluation of $\mathcal{N}$ near $U$ allow us to determine
$\kappa(\mathcal{N},U)$ with high probability. Specifically, we will make use
of a $p\kern 0.80002pt\text{-adic}$ point in spinor space
$(\eta^{(\epsilon)},\tilde{\eta}^{(\epsilon)})$ as constructed earlier in this
section to satisfy Eq. (135). The corresponding point $(\eta,\tilde{\eta})$ on
the variety $U$ can be thought of as any of the infinitely many $p\kern
0.80002pt\text{-adic}$ points on $U$ with the same first $p\kern
0.80002pt\text{-adic}$ digit as
$(\eta^{(\epsilon)},\tilde{\eta}^{(\epsilon)})$. We will argue that the
probability of evaluating $\mathcal{N}$ at the point
$(\eta^{(\epsilon)},\tilde{\eta}^{(\epsilon)})$ and finding that its $p\kern
0.80002pt\text{-adic}$ valuation exceeds $\kappa(\mathcal{N},U)$ is small. We
begin by noting that, by Eq. (145), $\mathcal{N}$ is an element of
$\mathfrak{m}^{\kappa(\mathcal{N},U)}_{(\eta,\tilde{\eta})}$. We can therefore
write $\mathcal{N}$ as

$\mathcal{N}=\sum_{|\beta|+|\tilde{\beta}|=\kappa(\mathcal{N},U)}n_{\beta,\tilde{\beta}}(\lambda,\tilde{\lambda})\prod_{i,\alpha}(\lambda_{i\alpha}-\eta_{i\alpha})^{\beta_{i\alpha}}\prod_{j,\dot{\alpha}}(\tilde{\lambda}_{j\dot{\alpha}}-\tilde{\eta}_{j\dot{\alpha}})^{\tilde{\beta}_{i\dot{\alpha}}}\,,$
(147)

where the summation runs over all sets of powers $\beta$ and $\tilde{\beta}$
such that the total degree of the product part of Eq. (147) is
$\kappa(\mathcal{N},U)$ and $n_{\beta,\tilde{\beta}}(\lambda,\tilde{\lambda})$
is a polynomial in the spinor variables. The indices $i,\,j$ run from $1$ to
$n$, with $n$ being the multiplicity of phase space, and
$\alpha,\,\dot{\alpha}$ are either $0$ or $1$. Evaluating $\mathcal{N}$ at
$(\eta^{(\epsilon)},\tilde{\eta}^{(\epsilon)})$ we obtain

$\mathcal{N}(\eta^{(\epsilon)},\tilde{\eta}^{(\epsilon)})=p^{\kappa(\mathcal{N},U)}\tilde{\mathcal{N}}(\eta^{(\epsilon)},\tilde{\eta}^{(\epsilon)})+\mathcal{O}\left(p^{\kappa(\mathcal{N},U)+1}\right)\,,$
(148)

where

$\tilde{\mathcal{N}}(\eta^{(\epsilon)},\tilde{\eta}^{(\epsilon)})=\sum_{|\beta|+|\tilde{\beta}|=\kappa(\mathcal{N},U)}n_{\beta,\tilde{\beta}}(\eta^{(\epsilon),0},\tilde{\eta}^{(\epsilon),0})\prod_{i,\alpha}\left(\eta_{i\alpha}^{(\epsilon),1}\right)^{\beta_{i\alpha}}\prod_{j,\dot{\alpha}}\left(\tilde{\eta}_{j\dot{\alpha}}^{{}^{(\epsilon),1}}\right)^{\tilde{\beta}_{i\dot{\alpha}}}.$
(149)

It is then clear that we can extract $\kappa(\mathcal{N},U)$ from the
numerical evaluation
$\mathcal{N}(\eta^{(\epsilon)},\tilde{\eta}^{(\epsilon)})$, if we can
understand the valuation of
$\tilde{\mathcal{N}}(\eta^{(\epsilon)},\tilde{\eta}^{(\epsilon)})$. We will
now argue that if $p$ is large, then with high probability

$\nu_{p}\left[\tilde{\mathcal{N}}(\eta^{(\epsilon)},\tilde{\eta}^{(\epsilon)})\right]=0\,.$
(150)

Firstly, we argue that there exists, with high probability, some
$n_{\beta,\tilde{\beta}}(\eta^{(\epsilon),0},\tilde{\eta}^{(\epsilon),0})$
that is not $\mathcal{O}(p)$. If all
$n_{\beta,\tilde{\beta}}(\eta^{(\epsilon),0},\tilde{\eta}^{(\epsilon),0})$
vanish modulo $p$, this would imply that
$\mathcal{N}\in\mathfrak{m}_{(\eta^{\prime},\tilde{\eta}^{\prime})}^{\kappa(\mathcal{N},U)+1}$,
for some point $(\eta^{\prime},\tilde{\eta}^{\prime})$ whose first $p\kern
0.80002pt\text{-adic}$ digit is given by
$(\eta^{(\epsilon),0},\tilde{\eta}^{(\epsilon),0})$. However, recalling the
discussion around Eq. (144), points such as
$(\eta^{\prime},\tilde{\eta}^{\prime})$ belong to higher codimension
varieties. As $(\eta^{(\epsilon),0},\tilde{\eta}^{(\epsilon),0})$ has been
chosen randomly and $p$ is large, such points are chosen with low probability.
Secondly, consider $\tilde{\mathcal{N}}$ as a polynomial in the
$(\eta^{(\epsilon),1},\tilde{\eta}^{(\epsilon),1})$ given fixed
$(\eta^{(\epsilon),0},\tilde{\eta}^{(\epsilon),0})$. The point
$(\eta^{(\epsilon),1},\tilde{\eta}^{(\epsilon),1})$ could then be close to a
zero of this polynomial. However, as this point was also chosen randomly and
$p$ is large, this also occurs with low probability as well.

In summary, for large $p$, given a point
$(\eta^{(\epsilon)},\tilde{\eta}^{(\epsilon)})$ close to $U$, and
$\mathcal{N}(\eta^{(\epsilon)},\tilde{\eta}^{(\epsilon)})$ we conclude that
with high probability

$\kappa(\mathcal{N},U)=\nu_{p}\left(\mathcal{N}(\eta^{(\epsilon)},\tilde{\eta}^{(\epsilon)})\right)\,.$
(151)

##  4 Ansatz Construction Algorithm

In this section, we leverage the technology described so far to build an
algorithm to construct Ansätze for rational functions in scattering
amplitudes. Specifically, for each coefficient $\mathcal{C}_{i}$ we discuss an
algorithm to construct a set of rational functions
$\\{\mathfrak{a}_{i,1},\ldots,\mathfrak{a}_{i,d_{i}}\\}$ of spinor variables
such that

$\mathcal{C}_{i}(\lambda,\tilde{\lambda})=\sum_{k=1}^{d_{i}}c_{i,k}\mathfrak{a}_{i,k}(\lambda,\tilde{\lambda})\,,$
(152)

where the $c_{i,k}$ are rational numbers. Importantly, this Ansatz has fewer
terms than those commonly considered in the literature based on functional
reconstruction techniques as it will take into account the analytical
properties of the rational functions. We consider the coefficients in least
common denominator form. That is,

$\mathcal{C}_{i}(\lambda,\tilde{\lambda})=\frac{\mathcal{N}_{i}(\lambda,\tilde{\lambda})}{\prod_{j=1}^{n_{i}}\mathcal{D}_{j}(\lambda,\tilde{\lambda})^{q_{ij}}}\,,$
(153)

where $\mathcal{N}_{i}$ and $\mathcal{D}_{j}$ are elements of
$\mathcal{R}_{n}$ and where $n_{i}$ is the number of distinct denominator
factors $\mathcal{D}_{j}$. We note that we allow the exponents $q_{ij}$ to be
negative, denoting numerator factors. For our procedure, we assume that the
set of denominator factors $\\{\mathcal{D}_{1},\ldots,\mathcal{D}_{n_{i}}\\}$
in Eq. (153) is known _a priori_. In physical applications, where the
transcendental functions are pure, it is conjectured that this set can be
constructed from the symbol alphabet Abreu:2018zmy . We further assume that we
can numerically evaluate the $\mathcal{C}_{i}$ $p\kern
0.80002pt\text{-adically}$, e.g. either from some analytic formula or from an
appropriate numerical algorithm.

###  4.1 Study of Singular Varieties

We begin by considering the behavior of the coefficient function
$\mathcal{C}_{i}$ on singular varieties. This procedure has two parts: first,
we find all relevant irreducible singular varieties; second, we perform
numerical evaluations near these varieties and interpret the result.

##### Analytic Study.

Let us consider the set of codimension-$m$ varieties on which the rational
functions in Eq. (153) may diverge. These are naturally associated to ideals
generated by the denominator factors in Eq. (153). Specifically, the set of
ideals that define the singular varieties at codimension $m$ is

$\mathfrak{D}^{(m)}=\Big{\\{}J\text{ such that
}J=\langle\mathcal{D}_{j_{1}},\ldots,\mathcal{D}_{j_{m}}\rangle_{R_{n}}\text{
and }\mathrm{codim}(J)=m\Big{\\}}\,,$ (154)

where the indices $j_{1},\ldots,j_{m}$ are all distinct and take values in
$1,\ldots,n_{i}$. We remind the reader that ideals generated by $m$ elements
are not necessarily of codimension $m$. The varieties associated to the ideals
in $\mathfrak{D}^{(m)}$ may be reducible. To this end, we consider the set of
irreducible varieties is given by

$\mathcal{V}^{(m)}=\Big{\\{}U\text{ such that }U=V(P)\text{ where