aainstitutetext: Dipartimento di Fisica e Astronomia, Università degli Studi
di Padova, Via Marzolo 8, I-35131 Padova, Italy.bbinstitutetext: INFN, Sezione
di Padova, Via Marzolo 8, I-35131 Padova, Italy.ccinstitutetext: Niels Bohr
International Academy, University of Copenhagen, Blegdamsvej 17, 2100
København Ø, Denmark

# Intersection Numbers from Higher-order Partial Differential Equations

Vsevolod Chestnov c Hjalte Frellesvig a,b Federico Gasparotto b Manoj K.
Mandal a,b
Pierpaolo Mastrolia
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
###### Abstract

We propose a new method for the evaluation of intersection numbers for twisted
meromorphic $n$-forms, through Stokes’ theorem in $n$ dimensions. It is based
on the solution of an $n$-th order partial differential equation and on the
evaluation of multivariate residues. We also present an algebraic expression
for the contribution from each multivariate residue. We illustrate our
approach with a number of simple examples from mathematics and physics.

##  1 Introduction

The intersection number of differential $n$-forms matsumoto1994 ;
matsumoto1998 ; OST2003 ; doi:10.1142/S0129167X13500948 ; goto2015 ; goto2015b
; Yoshiaki-GOTO2015203 ; Mizera:2017rqa ; matsubaraheo2019algorithm ;
Ohara98intersectionnumbers ; https://doi.org/10.48550/arxiv.2006.07848 ;
https://doi.org/10.48550/arxiv.2008.03176 ;
https://doi.org/10.48550/arxiv.2104.12584  acts like an inner product in the
de Rham twisted cohomology group, namely the quotient space of the closed
forms modulo the exact forms, and therefore, it yields the decomposition of
differential forms in a basis of forms, generating the vector space
Mastrolia:2018uzb ; Frellesvig:2019kgj ; Frellesvig:2019uqt ;
Frellesvig:2020qot .

A wide class of integral functions, such as Aomoto-Gel’fand integrals, Euler-
Mellin integrals, Gel’fand-Kapranov-Zelevinsky integrals, to name a few, which
embed and generalise Feynman integrals, can be considered as the pairing of
regulated integration domains, known as twisted cycles, and of regulated
forms, known as twisted cocycles aomoto2011theory . Within this definition,
the integrand appears as the product of a multivalued function, defined as
twist and of a differential form. The twist carries information on the
integral regularisation: for the case of dimensionally regulated Feynman
integrals, the space-time dimensionality appears in the exponent of the twist.
In this fashion, the algebraic properties of the integrals can be thought as
coming more fundamentally from the algebraic properties of the corresponding
cycles and cocycles. In particular, evaluation of intersection numbers for
twisted differential forms becomes a crucial operation to derive linear and
quadratic relations for integrals mentioned above, and to systematically
derive differential and difference equations the latter obey Mastrolia:2018uzb
; Frellesvig:2019kgj ; Frellesvig:2019uqt ; Frellesvig:2020qot ;
Mizera:2019vvs ; Chen:2020uyk ; Weinzierl:2020gda ; Caron-Huot:2021xqj ;
Caron-Huot:2021iev ; Chen:2022lzr . See Mizera:2019ose ; Cacciatori:2021nli ;
Abreu:2022mfk ; Weinzierl:2022eaz ; Mastrolia:2022tww ; Frellesvig:2021vem ;
Mandal:2022vok , for recent reviews, and Chen:2022lzr ; Ma:2021cxg  for
applications to multi-loop calculus. Applications of intersection theory and
co-homology to diagrammatic coaction have been presented in Abreu:2019wzk ;
Abreu:2019xep ; Abreu:2021vhb ; Abreu:2022erj , and to other interesting
physical contexts in Mizera:2017cqs ; Kaderli:2019dny ; Kalyanapuram:2020vil ;
Weinzierl:2020nhw .

The evaluation of the intersection numbers for twisted forms is based on the
twisted version of Stokes’ theorem cho1995 . In particular, for the case of
logarithmic $n$-forms, intersection numbers can be computed by applying the
algorithm proposed in matsumoto1998  or by means of the global residue theorem
Mizera:2017rqa . For generic meromorphic $n$-forms, the evaluation procedure
can become computationally more demanding, and it can be performed by means of
an iterative approach, as proposed in Mizera:2019gea ; Frellesvig:2019uqt ;
Frellesvig:2020qot , elaborating on Ohara98intersectionnumbers . The iterative
approach has been further refined in Weinzierl:2020xyy , by exploiting the
invariance of the intersection numbers for forms belonging to the same
cohomology classes. In Caron-Huot:2021xqj ; Caron-Huot:2021iev , this
algorithm has been extended to account also for the relative cohomology cases
matsumoto2018relative , required when dealing with singularities of the
integrand which are not regulated by the twist.

As an alternative to the evaluation procedure based on the Stokes’ theorem,
intersection numbers can also be computed by solving the secondary equation
built from the Pfaffian systems Matsubara-Heo-Takayama-2020b ;
matsubaraheo2019algorithm ; Chestnov:2022alh . Within this algorithm, the
determination of Pfaffian systems obeyed by the generators of the cohomology
group is required, and efficient methods for their derivation have recently
started to be proposed by means of Macaulay matrix in Chestnov:2022alh .

In this article, we propose a new algorithm for the computation of the
intersection number of twisted $n$-forms, based on a novel way of applying
Stokes’ theorem that requires the solution of a higher-order partial
differential equation and the application of the multivariate residues. The
computational algorithm hereby proposed can be considered as a natural
extension of the univariate case cho1995 , and just as the latter its
application requires the solution of a (partial) differential equation around
each intersection point, that belongs to the set of zeroes of the twist. In
this work, we show that the solution of the differential equation can be found
analytically by multiple Laurent series expansions, and that the each residues
admit a closed expressions, in terms of the Laurent coefficients of the two
forms entering the pairing and of the twist.

The structure of the paper is as follows: in Section 2 we discuss aspects of
twisted cohomology theory and intersection numbers, and we introduce a new
method for the computation of multivariate intersection numbers as
multivariate residues, introducing a higher-order partial differential
equation and discussing its solution, locally around each intersection point.
In Section 3 we give a closed, algebraic expression for each residue,
contributing to the multivariate intersection number. In Section 4 we
summarize how intersection numbers can be used to derive integral relations,
and Section 5 contains application of our new approach to integral functions
of interests for physics and mathematics. Section 6 contains our concluding
remarks. The paper includes four appendices: Appendix A contains the
derivation of the algebraic expression given in Section 3; Appendix B contains
a different approach to an algebraic expression for the individual
contributions to the intersection number. Appendix C contains further details
of the examples discussed in Section 5. Finally, Appendix D contains the link
of our new approach to Matsumoto’s algorithm, explicitly shown in the simple
case of 2-forms.

##  2 Intersection numbers for twisted n-forms

###  2.1 Twisted cohomology

Let $\mathcal{B}_{i}$, with $i=1,\ldots,m$ , be complex polynomials in the
variables $z=(z_{1},\ldots,z_{n})$. We introduce an oriented manifold
$X=\mathbb{C}^{n}-\bigcup_{i=1}^{m}{\mathcal{S}}_{i}$, where the hypersurfaces
${\mathcal{S}}_{i}$ are identified by the equations:

$\displaystyle\mathcal{S}_{i}:=\bigl{\\{}z\>|\>\mathcal{B}_{i}(z)=0\bigr{\\}}\
.$ (1)

We introduce the Aomoto-Gel’fand integrals, defined as twisted period
integrals,

$\displaystyle\int_{\Gamma^{(\mathbf{n})}}u\>\varphi^{(\mathbf{n})}\equiv\int_{\Gamma^{(\mathbf{n})}}u\>\hat{\varphi}^{(\mathbf{n})}\>\mathrm{d}z_{1}\wedge\ldots\wedge\mathrm{d}z_{n}\
,$ (2)

where: $u$ is a multivalued function called the twist, which regulates the
integral; $\Gamma^{(\mathbf{n})}$ is a regularised cycle called twisted or
loaded cycle, i.e. a $n$-chain with empty boundary on $X$ (usually
$\Gamma^{(\mathbf{n})}$ is denoted as
$\Gamma^{(\mathbf{n})}\equiv\Gamma^{(\mathbf{n})}\otimes u$ to separate the
integration domain $\Gamma^{(\mathbf{n})}$ and a specific choice of the branch
of multivalued $u$ along it); $\varphi^{(\mathbf{n})}$ is a meromorphic
differential $n$-form defined on $X$, called the twisted cocycle. In general
$u$ is a multivalued function that “vanishes” on the integration boundary:
$u(\partial{\Gamma^{(\mathbf{n})}})=0$. The latter property ensures that for
any generic $({\bf n-1})$-form $\varphi^{(\mathbf{n-1})}$ the integral of the
total differential is zero:

$\displaystyle
0=\int_{\Gamma^{(\mathbf{n})}}\mathrm{d}(u\,\varphi^{(\mathbf{n-1})})=\int_{\Gamma^{(\mathbf{n})}}u\,\nabla_{\omega}\,\varphi^{(\mathbf{n-1})}\
,$ (3)

where we introduced the covariant derivative:

$\displaystyle\nabla_{\omega}:=\mathrm{d}+\omega\wedge=u^{-1}\cdot\mathrm{d}\cdot
u\ ,\qquad{\rm
with}\qquad\omega\equiv\sum_{i=1}^{n}\hat{\omega}_{i}\>\mathrm{d}z_{i}=\mathrm{d}\log(u)\
,$ (4)

with $\hat{\omega}_{i}=u^{-1}\partial_{z_{i}}u$ .

Aomoto-Gel’fand integrals represent a wide class of special functions, such as
Gauss hypergeometric functions, Lauricella functions, and their
generalizations, Euler-type integrals, and Feynman integrals Mastrolia:2018uzb
. Integrals of this class are invariant under shifting of the differential
$n$-form by a covariant derivative:
$\varphi^{(\mathbf{n})}\to\varphi^{(\mathbf{n})}+\nabla_{\omega}\,\varphi^{(\mathbf{n-1})}$
, explicitly:

$\displaystyle\int_{\Gamma^{(\mathbf{n})}}u\,\varphi^{(\mathbf{n})}=\int_{\Gamma^{(\mathbf{n})}}u\,(\varphi^{(\mathbf{n})}+\nabla_{\omega}\,\varphi^{(\mathbf{n-1})})\
.$ (5)

Similar results are obtained also for the so called dual integrals, obtained
from the integrals (2) by replacing $u\to u^{-1}$ and $\omega\to-\omega$ in
definition (4).

In the case of Feynman integrals in Baikov representation Baikov:1996iu , the
twist $u$ admits the following factorized form:

$\displaystyle u=\prod_{i=1}^{m}{\mathcal{B}}_{i}^{\gamma_{i}}\ ,$ (6)

where the exponents $\gamma_{i}$ are non-integer, and the factors
$\mathcal{B}_{i}$ are polynomials build out of kinematics and Baikov variables
(corresponding to propagators). For this set of functions, analyticity,
unitarity, and algebraic structure are related to the geometry captured by the
Morse function $h:=\mathrm{Re}(\log(u))$, see Witten:2010cx ; Lee:2013hzt .

The multivalued twist $u$ carries information on the regularization: for
dimensionally regulated Feynman integrals, it depends on the integration
variables as well as on the external scales, such as the Mandelstam invariants
and masses (all appearing in the polynomials $\mathcal{B}_{i}$), and on the
space-time dimensionality $d$ (appearing in the $\gamma_{i}$). The topological
information of integrals and dual integrals are contained in $\omega$ that is
a differential form with zeroes and poles, collected in the respective sets:

$\displaystyle\mathbb{Z}_{\omega}=\bigl{\\{}\text{zeros of
$\omega$}\bigr{\\}}\ ,\quad\text{and}\quad\mathbb{P}_{\omega}=\\{{\rm poles\
of\ }\omega\\}\cup\\{\infty\\}\ .$ (7)

The invariance of integrals and dual integrals under the transformation (5)
can be used to expose the algebraic structure of Aomoto-Gel’fand integrals.
Let us introduce two vector spaces of twisted cocycles: the twisted $n$-th
cohomology group,

$\displaystyle H^{\mathbf{n}}_{\omega}$ $\displaystyle=\frac{{\rm
Ker}(\nabla_{\omega}:\varphi^{(\mathbf{n})}\to\varphi^{(\mathbf{n+1})})}{{\rm
Im}(\nabla_{\omega}:\varphi^{(\mathbf{n-1})}\to\varphi^{(\mathbf{n})})}\ ,$
(8)

which is the quotient space of closed $n$-forms $\\{\varphi^{(\mathbf{n})}\,|\
\nabla_{\omega}\varphi^{(\mathbf{n})}=0\\}$ modulo exact forms
$\\{\varphi^{(\mathbf{n})}\,|\
\varphi^{(\mathbf{n})}=\nabla_{\omega}{\varphi^{(\mathbf{n-1})}}\\}$; and the
dual twisted cohomology group:
$(H^{\mathbf{n}}_{\omega})^{\vee}:=H^{\mathbf{n}}_{-\omega}$ . These spaces
are isomorphic, and their dimension:

$\nu={\rm dim}(H_{\mathbf{n}}^{\pm\omega})\ ,$ (9)

can be determined by counting the number of critical points of
${\mathcal{B}_{i}}$ , namely $\nu={\rm dim}({\mathbb{Z}}_{\omega})$
Lee:2013hzt , or equivalently from the Euler characterisics
$\chi({\mathbb{P}}_{\omega})$ of the projective variety generated by the poles
of $\omega$, as $\nu=(-1)^{n}(n+1-\chi({\mathbb{P}}_{\omega}))$
Frellesvig:2019uqt , see also Bitoun:2017nre ; Agostini:2022cgv , or by the
Shape Lemma Frellesvig:2020qot .

We denote the elements of the twisted cohomology (8) as
$\langle\varphi_{L}|\in H^{\mathbf{n}}_{\omega}$, $|\varphi_{R}\rangle\in
H^{\mathbf{n}}_{-\omega}$, and use them to define the following natural
twisted Poincarè pairings:

  * •

Integrals:

$I=\int_{\Gamma^{(\mathbf{n})}}u\,\varphi_{L}\ ;$ (10)

  * •

Dual integrals:

${I^{\vee}}=\int_{\Gamma^{(\mathbf{n})}}u^{-1}\,\varphi_{R}\ ;$ (11)

  * •

Intersection numbers for twisted cocycles

$\langle\varphi_{L}|\varphi_{R}\rangle\equiv(2\pi\mathrm{i})^{-n}\int_{X}(u\,\varphi_{L})\wedge(u^{-1}\,\varphi_{R})$
(12)

where $\varphi_{L,R}$ are understood to have compact support on $X$.

###  2.2 Partial Differential Equation

By elaborating on the method proposed in matsumoto1998 , we hereby propose to
evaluate the intersection number for n-forms, using the multivariate Stokes
theorem, yielding

$\displaystyle\langle\varphi_{L}^{({\bf n})}\>|\>\varphi_{R}^{({\bf
n})}\rangle$ $\displaystyle=$ $\displaystyle\sum_{p\in\mathbb{P}_{\omega}}{\rm
Res}_{z=p}(\psi\,\varphi_{R}^{({\bf n})})\ ,$ (13)

where:

  * •

$\psi$ is a function (0-form), that obeys the following $n$-th order partial
differential equation ($n$PDE),

$\displaystyle\frac{\partial^{n}}{\partial z_{n}\,\partial
z_{n-1}\ldots\partial{z_{1}}}(u\,\psi)$
$\displaystyle=u\,\hat{\varphi}_{L}^{({\bf n})}\ .$ (14)

  * •

$p=(p_{1},p_{2},\ldots,p_{n})\in\mathbb{P}_{\omega}$ is a pole of $\omega$,
i.e. an intersection point of singular hypersurface $\mathcal{S}_{i}$ defined
in eq. (1), at finite location or at infinity.

  * •

The residue symbol stands for

$\displaystyle{\rm Res}_{z=p}(f)={\rm Res}_{z_{n}={p_{n}}}\ldots{\rm
Res}_{z_{1}={p_{1}}}(f)=(2\pi\mathrm{i})^{-n}\int_{\circlearrowleft_{1}\wedge\ldots\wedge\circlearrowleft_{n}}f\>\mathrm{d}z_{1}\wedge\ldots\wedge\mathrm{d}z_{n}\
,$ (15)

where the integral goes over a product of small loops $\circlearrowleft_{i}$ ,
each encircling the corresponding pole $z_{i}=p_{i}$ in the $z_{i}$-plane, see
for example Sogaard:2013fpa .

Equation (13) can be derived by rewriting the intersection number as a flux,

$\displaystyle\int_{X}(u\,\varphi_{L}^{({\bf
n})})\wedge(u^{-1}\varphi_{R}^{({\bf
n})})=\int_{X}\partial_{n}\ldots\partial_{1}\eta=\int_{\partial X}\eta$ (16)

owing to the existence (locally, around each pole) of the differential ${\bf
n}$-form $\eta$, defined as,

$\displaystyle\eta:=(u\,\psi)\,(u^{-1}\,\varphi_{R}^{({\bf
n})})=\psi\,\varphi_{R}^{({\bf n})}\ .$ (17)

The last multifold integration over the boundary $\partial X$ is carried out
by multivariate residue theorem, along the lines of matsumoto1998 . In fact,
by requiring that $\psi$ obeys eq. (14), the $\eta$ obtains the desired
property:

$\displaystyle\partial_{n}\ldots\partial_{1}\eta=(u\,\varphi_{L}^{({\bf
n})})\wedge(u^{-1}\varphi_{R}^{({\bf n})})\ ,$ (18)

because $\varphi_{R}^{({\bf n})}$ is closed w.r.t. all variables, namely:

$\displaystyle\partial_{n}\ldots\partial_{1}(u^{-1}\varphi_{R}^{({\bf
n})})=u^{-1}\nabla_{-\omega_{n}}\nabla_{-\omega_{n-1}}\ldots\nabla_{-\omega_{1}}\varphi_{R}^{({\bf
n})}=0\ .$ (19)

Equation (13) has already appeared in matsumoto1998 , while eq. (14) is new
and constitutes the main result of this communication, as it offers a new
algorithm for the determination of $\psi$, alternative to the one presented in
matsumoto1998 , hence a simpler strategy for the evaluation of the
intersection numbers.

Equation (14) can be also written as,

$\displaystyle\partial_{n}\partial_{n-1}\ldots\partial_{1}(u\,\psi)$
$\displaystyle=$ $\displaystyle
u\,\nabla_{\omega_{n}}\nabla_{\omega_{n-1}}\ldots\nabla_{\omega_{1}}\psi=u\,\varphi_{L}^{({\bf
n})}\ ,$ (20)

finally yielding,

$\displaystyle\boxed{\nabla_{\omega_{n}}\nabla_{\omega_{n-1}}\ldots\nabla_{\omega_{1}}\psi=\varphi_{L}^{({\bf
n})}}\ .$ (21)

We observe that this $n$PDE, equivalent to eq. (14), is the natural extension
of the univariate equation presented in cho1995  for the single variable case,
and it constitutes the first main result of this study.

###  2.3 Solution

The solution of eqs. (14, 21) can be formally written as 111 In ref.
matsumoto2018relative , the solution $\psi$ for the twisted case with
regulated pole is written by considering a modified integration contour,
accounting for the contribution of monodromy. It can be shown that, around
each (regulated) singular point, it is equivalent to the one considered here.
:

$\displaystyle\psi=u^{-1}\int_{z_{0}}^{z}u\,\varphi_{L}^{({\bf n})}\ .$ (22)

A crucial observation is in order. The global solution $\psi$ is, in general,
a transcendental function. The evaluation of the intersection number in (13),
though, because of the calculation of residues, requires the knowledge of
$\psi$ only locally around each of the contributing pole, say
$\psi_{p}=\psi|_{z\to p}$ . To determine the local expression of $\psi$ around
the point $p$, we propose two equivalent strategies:

  1. 1.

solution by ansatz 222 Here and in the following, we adopt the multivariate
exponent notation:
$(z-p)^{\boldsymbol{a}}\equiv(z_{1}-p_{1})^{a_{1}}\ldots(z_{n}-p_{n})^{a_{n}}$
. :

$\displaystyle\psi_{p}\equiv\psi|_{z\to
p}=\sum_{{\boldsymbol{a}}={\boldsymbol{a}}_{\rm min}}^{{\boldsymbol{a}}_{\rm
max}}\psi_{p,{\boldsymbol{a}}}\,(z-p)^{\boldsymbol{a}}\ ,$ (23)

whose coefficients $\psi_{p,{\boldsymbol{a}}}$ can be determined by requiring
the fulfillment of (21) around the pole $p$. The values of the
${\boldsymbol{a}}_{\rm min}$ and the ${\boldsymbol{a}}_{\rm max}$ depend on
the Laurent expansion of $\varphi_{L}^{({\bf n})}$ and $\varphi_{R}^{({\bf
n})}$ around $p$. The determination of the Laurent coefficients
$\psi_{p,{\boldsymbol{a}}}$ can be carried out by solving the triangular
systems of linear equations that arises after inserting the ansatz in the
multivariate differential equation.

  2. 2.

solution by series expansion:

$\displaystyle\psi_{p}\equiv\psi\Big{|}_{z\to
p}=\bigl{(}u^{-1}\bigr{)}\Big{|}_{z\to
p}\int^{z}_{p}\bigl{(}u\varphi_{L}^{(\mathbf{n})}\bigr{)}\Big{|}_{z\to p}\ ,$
(24)

which can be directly obtained by a series expansion under the integral sign,
and a subsequent multifold integration. Taking $p=0$ without loss of
generality, we observe that as $z\to 0$, the twist (6) admits a factorized
expansion $u|_{z\to
0}=z^{\gamma^{\prime}}\cdot\sum_{{\boldsymbol{i}}\geq{\boldsymbol{0}}}u_{{\boldsymbol{i}}}z^{{\boldsymbol{i}}}$
with non-integer exponents $\gamma^{\prime}$. A similar expansion holds for
the cocycles: $\hat{\varphi}_{L}|_{z\to
0}=\sum_{{\boldsymbol{i}}\geq{\boldsymbol{\mu_{L}}}}\varphi_{L,{\boldsymbol{i}}}\>z^{{\boldsymbol{i}}}$
and $\hat{\varphi}_{R}|_{z\to
0}=\sum_{{\boldsymbol{i}}\geq{\boldsymbol{\mu_{R}}}}\varphi_{R,{\boldsymbol{i}}}\>z^{{\boldsymbol{i}}}$
with integer leading exponents
${\boldsymbol{\mu_{L}}},{\boldsymbol{\mu_{R}}}\in\mathbb{Z}^{n}$. Using these
expansions integration in eq. (24) can be done term by term via
$\int^{z}\mathrm{d}x\>x^{a}=z^{a+1}/(a+1)$.

Let us remark that the second strategy allows for a straightforward
determination of the intersection numbers bypassing any linear system solving
procedure, and it constitutes the second main result of this study.

###  2.4 Simplified formulas from Rescaling

In the previous sections we saw how the intersection number was given by

$\displaystyle\langle\varphi_{L}|\varphi_{R}\rangle=\sum_{p\in\mathbb{P}_{\omega}}{\rm
Res}_{z=p}(\psi\varphi_{R})\ ,\qquad\text{where formally}\qquad\psi=u^{-1}\int
u\,\varphi_{L}\ .$ (25)

It is not hard to see that the above expressions are invariant under the
following simultaneous rescalings:

$\displaystyle u\rightarrow
u/q\,,\qquad\varphi_{L}\rightarrow\varphi_{L}q\,,\qquad\varphi_{R}\rightarrow\varphi_{R}/q\,,\qquad\psi\rightarrow\psi
q\,,$ (26)

where $q$ may be any 333 To be precise, the function $q$ here has to be
meromorphic and have all poles regulated by $u$.  function of $z$. Since each
individual residue of eq. (25) possesses this invariance, the rescaling may be
done locally at each individual $p$. Such rescalings are of interest since
certain specific choices for $q$ introduce simplifications. We will look at
two in particular:

##### Right rescaling.

This is defined as the choice $q=\hat{\varphi}_{R}$. It corresponds to the
rescalings

$\displaystyle u\rightarrow
u_{R}:=u/\hat{\varphi}_{R}\,,\qquad\varphi_{L}\rightarrow\phi:=\varphi_{L}\hat{\varphi}_{R}\,,\qquad\hat{\varphi}_{R}\rightarrow
1\,,\qquad\psi\rightarrow f:=\psi\hat{\varphi}_{R}\,,$ (27)

and we also define $\omega_{R}:=d\log(u_{R})$. The advantage of this rescaling
is that the residues appearing in eq. (25) become $\text{Res}(f)$ making it
extremely clear to which order in $z$ the differential equation (21), which
now reads

$\displaystyle\nabla_{\omega_{R},1}\cdots\nabla_{\omega_{R},n}f=\phi\,,$ (28)

has to be solved (i.e. all terms of $\mathcal{O}(z_{i}^{0})$ can be ignored).
It is this form of eq. (21) which is the starting point of the algebraic
solution given in Section 3.

##### Left rescaling

This form of rescaling is defined by the choice $q=1/\hat{\varphi}_{L}$. It
corresponds to

$\displaystyle u\rightarrow
u_{L}:=u\hat{\varphi}_{L}\,,\qquad\hat{\varphi}_{L}\rightarrow
1\,,\qquad\varphi_{R}\rightarrow\phi:=\hat{\varphi}_{L}\varphi_{R}\,,\qquad\psi\rightarrow
g:=\psi/\hat{\varphi}_{L}\,,$ (29)

and we also define $\omega_{L}:=d\log(u_{L})$. This rescaling has the property
that eq. (21) becomes

$\displaystyle\hat{\nabla}_{\omega_{L},1}\cdots\hat{\nabla}_{\omega_{L},n}\,g=1\,,$
(30)

which is conceptually simpler to solve than the case with a generic function
on the RHS.

Finally let us remark, that another source of simplifications is the shift
invariance of intersection numbers:

$\displaystyle\langle\varphi_{L}\>|\>\varphi_{R}\rangle=\langle\varphi_{L}+\nabla_{\omega}\xi_{L}\>|\>\varphi_{R}+\nabla_{-\omega}\xi_{R}\rangle\
,$ (31)

which is valid for generic $(n-1)$-forms $\xi_{L}$ and $\xi_{R}$, but
discussion of this falls out of our scope. Now we move on to applications of
the formalism that was introduced here.

##  3 Algebraic solutions

In this section we will introduce an algebraic solution for the contribution
to the intersection number coming from each individual residue. Our main
formula of this section (36) is derived by solving the $n$PDE using an ansatz,
and by optimizing the algebraic manipulations with the help of vector
compositions andrews_1984 .

###  3.1 Algebraic expression for residues

In Section 2 we saw that multivariate intersection numbers can be expressed as
a sum over contributions from a number of points, with each contribution
expressed as a residue. In the “right rescaling” framework of Section 2.4 the
relation is given as

$\displaystyle\langle\varphi_{L}|\varphi_{R}\rangle=\sum_{p\in\mathbb{P}_{\omega}}\text{Res}_{z=p}(f)$
(32)

where $f$ is defined as the solution to

$\displaystyle\nabla_{\omega_{R,1}}\cdots\nabla_{\omega_{R,n}}(f)=\varphi_{L}\hat{\varphi}_{R}$
(33)

where

$\displaystyle\nabla_{\omega_{R,i}}:=(\partial_{z_{i}}+(\partial_{z_{i}}w))\,\text{d}z_{i}\qquad\qquad\text{with}\qquad\qquad
w:=\log(u/\hat{\varphi}_{R})$ (34)

In Section 2.3 it was discussed how to solve systems of the form of eq. (33)
by making a series ansatz for $f$ and solving it locally around each point. In
this section we are going to pursue that direction a step further, to obtain
an algebraic solution for each term in eq. (32). The first step is to write
$w$ and $\hat{\varphi}_{L}\hat{\varphi}_{R}$ as series around each point,
where the appropriate variable changes have been made in order to ensure that
the point is located at $\boldsymbol{z}=\boldsymbol{0}$ and in addition that
the series expansions are well defined so no further variable changes (or
blowups) are needed. With this we may write

$\displaystyle\hat{\varphi}_{L}\hat{\varphi}_{R}$
$\displaystyle=\sum_{\boldsymbol{i}=\boldsymbol{\mu}_{\boldsymbol{\phi}}\\!\\!\\!}^{\boldsymbol{\infty}}\phi_{\boldsymbol{i}}z^{\boldsymbol{i}}\qquad\quad\text{and}\qquad\quad
w=\log(u/\hat{\varphi}_{R})=\sum_{\boldsymbol{i}=\boldsymbol{0}}^{\boldsymbol{\infty}}w_{\boldsymbol{i}}z^{\boldsymbol{i}}\;+\;\sum_{j=1}^{n}v_{j}\log(z_{j})$
(35)

where $z^{\boldsymbol{i}}:=z_{1}^{i_{1}}\cdots z_{n}^{i_{n}}$. In this
notation $\boldsymbol{\mu}_{\boldsymbol{\phi}}$ should be interpreted as the
powers corresponding to the leading term in the expansion of
$\hat{\varphi}_{L}\hat{\varphi}_{R}$. The $w$-expansion444One might consider
why $w$ has these log terms. Clearly such terms will appear if $u$ has a
factor of $z^{\gamma}$, and in addition the requirement is that the
derivatives of $w$ are Laurent polynomials, not that $w$ itself is. This is
what rules out terms of the form $z_{1}\log(z_{1})$ or $z_{2}\log(z_{1})$ as
such terms do not have all derivatives being Laurent. This is also why the
branch choice of the logarithms is irrelevant, since potential factors of
$2\pi i$ will vanish when taking the derivatives. begins at powers
$\boldsymbol{0}$ as lower ones cannot appear when $u$ is of the form
$\prod\mathcal{B}^{\gamma}$. Using an ansazt for $f$ as a Laurent series
expansion around each intersection point, the $n$PDE constraints the
coefficients555 This technique of solving differential equations is also known
as the “Frobenius method”. , so that the residue of $f$ can be obtained in
closed form as,

$\displaystyle\boxed{\,\text{Res}(f)\,=\\!\sum_{\boldsymbol{\tau}=\boldsymbol{0}}^{\\!{-}\boldsymbol{\mu}_{\boldsymbol{\phi}}{-}\boldsymbol{2}}\\!\sum_{\sigma\in\text{VC}(\boldsymbol{\tau})}\\!\left(\frac{(-1)^{|\sigma|}}{U(\boldsymbol{0},\boldsymbol{0})}\prod_{i=1}^{|\sigma|}\frac{U(\boldsymbol{\tau}{-}\sum_{j<i}\\!\boldsymbol{\sigma}_{\boldsymbol{j}},\boldsymbol{\sigma}_{\boldsymbol{i}})}{U(\boldsymbol{\tau}{-}\sum_{j<i}\\!\boldsymbol{\sigma}_{\boldsymbol{j}},\boldsymbol{0})}\right)\\!\phi_{(-\boldsymbol{\tau}-\boldsymbol{2})}\,}$
(36)

This formula constitutes the third main result of this work, and more details
on its derivation can be found in Appendix A.

The symbol VC is short for vector compositions, which are a known operation in
combinatorics, that results in a set of lists of $n$-vectors, with $|\sigma|$
referring to the lengths of those lists. For the full explanation of vector
compositions see Section 3.2. Additionally $U$ is a function dependent only on
the terms in the expansion of $w$. We will define it in terms of a different
function $R$ as

$\displaystyle U(\boldsymbol{\eta},\boldsymbol{\lambda})$
$\displaystyle:=R(\boldsymbol{\lambda},\,{-}\boldsymbol{\eta}{-}\boldsymbol{1})$
(37)

$R$ is defined in its full generality in Appendix A.1.2. Here we will limit
ourselves to writing expressions for $R$ for certain small values of $n$:

$\displaystyle n=1:\quad$ $\displaystyle
R(\alpha,\beta)\,=\,(\beta{+}v)\delta_{\alpha,0}+\alpha w_{\alpha}$ (38)
$\displaystyle n=2:$ $\displaystyle
R(\boldsymbol{\alpha},\boldsymbol{\beta})=(\beta_{1}{+}v_{1})(\beta_{2}{+}v_{2})\delta_{\boldsymbol{\alpha},\boldsymbol{0}}+\big{(}\alpha_{1}\alpha_{2}+\alpha_{1}(\beta_{2}{+}v_{2})+\alpha_{2}(\beta_{1}{+}v_{1})\big{)}w_{\boldsymbol{\alpha}}$
$\displaystyle\>+\sum_{\boldsymbol{j}\,=\,\boldsymbol{0}}^{\boldsymbol{\alpha}}(\alpha_{1}{-}j_{1})j_{2}\,w_{\boldsymbol{\alpha}-\boldsymbol{j}}w_{\boldsymbol{j}}$
(39) $\displaystyle n=3:\quad$ $\displaystyle
R(\boldsymbol{\alpha},\boldsymbol{\beta})=(\beta_{1}{+}v_{1})(\beta_{2}{+}v_{2})(\beta_{3}{+}v_{3})\delta_{\boldsymbol{\alpha},\boldsymbol{0}}$
$\displaystyle\>+\big{(}(\alpha_{1}{+}\beta_{1}{+}v_{1})(\alpha_{2}{+}\beta_{2}{+}v_{2})(\alpha_{3}{+}\beta_{3}{+}v_{3})-(\beta_{1}{+}v_{1})(\beta_{2}{+}v_{2})(\beta_{3}{+}v_{3})\big{)}w_{\boldsymbol{\alpha}}$
$\displaystyle\>+\sum_{\boldsymbol{j}\,=\,\boldsymbol{0}}^{\boldsymbol{\alpha}}\big{(}(\alpha_{1}{-}j_{1})j_{2}(j_{3}{+}\beta_{3}{+}v_{3})+(\alpha_{2}{-}j_{2})j_{3}(j_{1}{+}\beta_{1}{+}v_{1})+(\alpha_{3}{-}j_{3})j_{1}(j_{2}{+}\beta_{2}{+}v_{2})\big{)}w_{\boldsymbol{\alpha}{-}\boldsymbol{j}}w_{\boldsymbol{j}}$
$\displaystyle\>+\sum_{\boldsymbol{j}=\boldsymbol{0}}^{\boldsymbol{\alpha}}\,\sum_{\boldsymbol{l}=\boldsymbol{0}}^{\boldsymbol{\alpha}{-}\boldsymbol{j}}\,(\alpha_{1}{-}j_{1}{-}l_{1})\,j_{2}\,l_{3}\,w_{\boldsymbol{\alpha}{-}\boldsymbol{j}{-}\boldsymbol{l}}\,w_{\boldsymbol{j}}\,w_{\boldsymbol{l}}$
(40)

For a general $n$-variate expression for $R$, see Appendix A.1.2.

###  3.2 Vector Compositions

In this section we will define and discuss the vector compositions
andrews_1984  that are playing an essential role in the algebraic expression
eq. (36).

$\displaystyle
S=\text{VC}(\boldsymbol{\tau})\qquad\text{with}\qquad\boldsymbol{\tau}=(\tau_{1},\ldots,\tau_{n})$
(41)

is a function of $n$ non-negative integers $\tau_{1}$ to $\tau_{n}$. Its
output $S$ is a set of ordered lists $\sigma$ each containing a number of
$n$-vectors $\boldsymbol{s}$. Those vectors contain non-negative integers and
the zero-vector is excluded, corresponding to all
$\boldsymbol{s}>\boldsymbol{0}$, where we order integer vectors using the
product order, see Appendix A.4 for a review . Interpreting each
$\boldsymbol{s}$ as a step in $\mathbb{Z}^{n}$, each $\sigma$ should
correspond to a path going from $\boldsymbol{0}$ to $\boldsymbol{\tau}$
implying the property
$\sum_{\boldsymbol{s}\in\sigma}\boldsymbol{s}=\boldsymbol{\tau}$. With this,
$S$ is defined as the set of all such paths. For more666Also note that vector
compositions are a variant of what is known in combinatorics as “lattice
paths”. on vector compositions, see e.g. andrews_1984 .

As an example, let us consider the case $n=2$ and $\boldsymbol{\tau}=(1,2)$,
with

Figure 1:  Paths appearing in eq. (42).

$\displaystyle\text{VC}(\boldsymbol{\tau})=\Big{\\{}$
$\displaystyle\left(\tbinom{1}{2}\right),$
$\displaystyle\left(\tbinom{1}{0},\tbinom{0}{2}\right),\left(\tbinom{1}{1},\tbinom{0}{1}\right),\left(\tbinom{0}{1},\tbinom{1}{1}\right),\left(\tbinom{0}{2},\tbinom{1}{0}\right),$
(42)
$\displaystyle\left(\tbinom{1}{0},\tbinom{0}{1},\tbinom{0}{1}\right),\left(\tbinom{0}{1},\tbinom{1}{0},\tbinom{0}{1}\right),\left(\tbinom{0}{1},\tbinom{0}{1},\tbinom{1}{0}\right)\Big{\\}}$

where the three rows correspond to entries with $|\sigma|$ being 1, 2, and 3,
respectively. This example is illustrated in Figure 1 as paths in
$\mathbb{Z}^{2}$.

We notice that in the $n=1$ case, the vector compositions reduce to standard
compositions, that is $\text{VC}(a)=\text{comp}(a)$ as long as we identify the
1-vectors as scalars. Compositions are a known operation from combinatorics
which describe the number of (ordered) ways an integer can be expressed as a
sum of positive integers, with an example being

$\displaystyle\text{comp}(4)=\big{\\{}(4),(3,1),(2,2),(1,3),(2,1,1),(1,2,1),(1,1,2),(1,1,1,1)\big{\\}}$
(43)

With this we wrap up our discussion/derivation of the algebraic expression for
the residue (36) and point the interested reader to Appendix A for further
details. Before we turn to examples of our new method, let us recall how
intersection numbers make their appearance in the physics of Feynman
integrals.

##  4 Integrals and Relations

Here we briefly review some practical applications of the twisted cohomology
theory to the study of Feynman integrals, focusing on the IBP decomposition
method.

Consider the bases generating the cohomology groups introduced in Section 2:
$\\{\langle e_{i}|\\}_{i=1,\ldots,\nu}\in H^{n}_{\omega}$ and
$\\{|h_{i}\rangle\\}_{i=1,\ldots,\nu}\in H^{n}_{-\omega}$ . These two bases
can be used to express the identity operator in the cohomology space
Mastrolia:2018uzb ; Frellesvig:2019kgj  as follows:

$\displaystyle\mathbb{I}_{c}=\sum_{i,j=1}^{\nu}|h_{i}\rangle\left(\mathbf{C}^{-1}\right)_{ij}\langle
e_{j}|\ ,$ (44)

where we defined the _metric matrix_

$\mathbf{C}_{ij}:=\langle e_{i}|h_{j}\rangle\ ,$ (45)

whose elements are intersection numbers of the twisted basis forms.

Linear relations for Aomoto-Gel’fand-Feynman integrals, as well as the
differential equations and the finite difference equation they obey are a
consequence of the purely algebraic application of the identity operators
defined above Mastrolia:2018uzb .

In the context of Feynman integral calculus, the decomposition of scattering
amplitudes in terms of master integrals (MIs), as well as the equations obeyed
by the latter, are derived by means of IBPs Tkachov:1981wb ; Chetyrkin:1981qh
and of the Laporta method Laporta:2001dd . Let us see now how these relations
emerge by employing the algebraic properties of twisted cocycles.

  * •

Decomposition of differential forms. Generic twisted cocycles and dual twisted
cocycles can be projected onto the bases in the correspsonding vector spaces
as,

$\displaystyle\langle\varphi_{L}|$ $\displaystyle=$
$\displaystyle\langle\varphi_{L}|\mathbb{I}_{c}=\sum_{i=1}^{\nu}c_{i}\,\langle
e_{i}|\ ,\quad{\rm with\ }\qquad
c_{i}=\sum_{j=1}^{\nu}\,\langle\varphi_{L}|h_{j}\rangle\,\left(\mathbf{C}^{-1}\right)_{ji}\
;$ (46) $\displaystyle|\varphi_{R}\rangle$ $\displaystyle=$
$\displaystyle\mathbb{I}_{c}|\varphi_{R}\rangle=\sum_{i=1}^{\nu}{c}_{i}^{\vee}\,|h_{i}\rangle\
,\quad{\rm with\
}\qquad{c}_{i}^{\vee}=\sum_{j=1}^{\nu}\,\left(\mathbf{C}^{-1}\right)_{ij}\,\langle
e_{j}|\varphi_{R}\rangle\ .$ (47)

The latter two formulas, dubbed _master decomposition formulas_ for (dual)
twisted cocycles Mastrolia:2018uzb ; Frellesvig:2019kgj , imply that the
decomposition of any (dual) Aomoto-Gel’fand-Feynman integral as a linear
combination of (dual) master integrals is an algebraic operation (similarly to
the decomposition/projection of any vector within a vector space,) which can
be carried out by computing intersection numbers of twisted de Rham
differential forms.

  * •

Integral decomposition. By using the master decomposition formulas of forms
and dual forms, integrals and dual integrals can be straightforwardly written
as,

$\displaystyle I=\sum_{i=1}^{\nu}c_{i}\,J_{i}\ ,\qquad{\rm
and}\qquad{I}^{\vee}=\sum_{i=1}^{\nu}{c}_{i}^{\vee}\,{J}_{i}^{\vee}\ ,$ (48)

respectively in terms the MIs $J_{i}$ , and of the dual MIs $J_{i}^{\vee}$ ,
for $i=1,\ldots,\nu$.

Finally let us remark, that the metric matrix (45), in general, differs from
the identity matrix. The Gram-Schmidt algorithm can be employed to build
orthonormal bases from generic sets of independent elements, using the
intersection numbers as scalar products. But more generally the coefficients
appearing in the formulas (46, 48) are independent of the respective dual
elements. Therefore, exploiting this freedom in choosing the corresponding
dual bases may yield striking simplifications Frellesvig:2019kgj ; Caron-
Huot:2021xqj ; Caron-Huot:2021iev . Note, that the decomposition formulas hold
also in the case of the relative twisted de Rham cohomology, which allows for
a relaxation of the non-integer condition for the exponents $\gamma_{i}$ that
appear in eq. (6), see matsumoto2018relative ; Caron-Huot:2021xqj ; Caron-
Huot:2021iev .

##  5 Examples

In this section we provide several examples of the multivariate differential
equation method for computation of intersection numbers. In Section 5.1 we
review some basic facts about the complex projective space
$\mathbb{C}\mathrm{P}^{n}$ that hosts the integration domain $X$. After that
we show three examples of $n=2$ variable intersection number calculations with
twists $u$ of various complexity. Explicit results presented below agree with
the iterative method of Mizera:2019gea ; Frellesvig:2019uqt ;
Frellesvig:2020qot .

###  5.1 Projective plane

A natural choice for the ambient space of the integration domain $X$
introduced in eq. (12) is the complex projective space
$\mathbb{C}\mathrm{P}^{n}$, which in $n=2$ dimensions is called the complex
projective plane $\mathbb{C}\mathrm{P}^{2}$.

Consider a triple of complex variables
$(Z_{1},Z_{2},Z_{3})\in\mathbb{C}^{3}\setminus(0,0,0)$ . The projective plane
$\mathbb{C}\mathrm{P}^{2}$ consists of equivalence classes of these triples
under a uniform rescaling: $(Z_{1},Z_{2},Z_{3})\equiv(\lambda Z_{1},\lambda
Z_{2},\lambda Z_{3})$ for $\lambda\in\mathbb{C}^{\ast}$. In other words,
$\mathbb{C}\mathrm{P}^{2}$ is the space of complex lines in $\mathbb{C}^{3}$
passing through the origin.

We turn $\mathbb{C}\mathrm{P}^{2}$ into a manifold by covering it with three
coordinate charts defined as follows:

$\displaystyle\\{U_{z},U_{y},U_{x}\\}_{i}:=\bigl{\\{}(Z_{1},Z_{2},Z_{3})\>\big{|}\>Z_{i}\neq
0\bigr{\\}}_{i\in\\{1,2,3\\}}\ ,$ (49)

that is, the sets of triples with one non-zero coordinate $Z_{i}$ , which we
can safely rescale away. Locally in these charts $\mathbb{C}\mathrm{P}^{2}$
looks like a copy of $\mathbb{C}^{2}$, so we introduce three pairs of local
coordinates shown in Table 1.

In the examples below we determine the distinct intersection points of various
curves $\mathcal{S}_{i}$ (see above eq. (2)) inside of
$\mathbb{C}\mathrm{P}^{2}$. Therefore it is important to understand how the
charts overlap with one another in order to avoid overcounting of points. This
can be easily deduced from the definition (49): for example, the $U_{z}\cap
U_{y}$ is given by triples with $Z_{1}\neq 0$ and $Z_{2}\neq 0$, while
$U_{y}\setminus U_{z}$ is determined by $Z_{1}=0$ and $Z_{2}\neq 0$.

Thus in the following we will work mainly in the $U_{z}$ chart along with the
corresponding line at infinity: the projective line $\mathbb{C}\mathrm{P}^{1}$
given by $Z_{1}=0$ . This line passes through the other two charts $U_{x}$ and
$U_{y}$ , where it is determined by $x_{2}=0$ and $y_{1}=0$ respectively. Now
that the stage is set, let us turn to the examples coming from the Feynman
integral calculus.

Chart | Local coordinates | Back to $U_{z}$
---|---|---
$U_{z}$ | $(1,z_{1},z_{2}):=(1,\tfrac{Z_{2}}{Z_{1}},\tfrac{Z_{3}}{Z_{1}})$ |
$U_{x}$ | $(x_{2},x_{1},1):=(\tfrac{Z_{1}}{Z_{3}},\tfrac{Z_{2}}{Z_{3}},1)$ | $(x_{1},x_{2})=(\tfrac{z_{1}}{z_{2}},\tfrac{1}{z_{2}})$
$U_{y}$ | $(y_{1},1,y_{2}):=(\tfrac{Z_{1}}{Z_{2}},1,\tfrac{Z_{3}}{Z_{2}})$ | $(y_{1},y_{2})=(\tfrac{1}{z_{1}},\tfrac{z_{2}}{z_{1}})$
Table 1:  The three coordinate charts (left) that cover
$\mathbb{C}\mathrm{P}^{2}$ with their local coordinates (middle). On the right
we show the coordinate transformations from the $U_{y}$ and $U_{x}$ charts
back to $U_{z}$. In this section we focus on the $U_{z}$ chart and its line at
infinity, i.e. the projective line $\mathbb{C}\mathrm{P}^{1}$ given by
$Z_{1}=0$.

###  5.2 Two-loop massless sunrise diagram

The first example is the massless sunrise diagram on the maximal cut. Using
the Baikov parametrization, we denote the two irreducible scalar products as
$z_{1}$ and $z_{2}$ . The twist $u$ is built from the Baikov polynomial on the
maximal cut. In the three coordinate charts of the projective plane
$\mathbb{C}\mathrm{P}^{2}$ it looks like this:

$\displaystyle
u=\begin{cases}z_{1}^{\gamma_{1}}z_{2}^{\gamma_{2}}\>(z_{1}+z_{2}-s)^{\gamma_{3}}&\text{in
$U_{z}$}\ ,\\\
x_{1}^{\gamma_{1}}x_{2}^{-\gamma_{0}}\>(1+x_{1}-sx_{2})^{\gamma_{3}}&\text{in
$U_{x}$}\ ,\\\
y_{1}^{-\gamma_{0}}y_{2}^{\gamma_{2}}\>(1-sy_{1}+y_{2})^{\gamma_{3}}&\text{in
$U_{y}$}\ ,\end{cases}$ (50)

where $\gamma_{0}:=\gamma_{1}+\gamma_{2}+\gamma_{3}$ , and the polynomial
factors (6) are
$\\{\mathcal{B}_{1},\mathcal{B}_{2},\mathcal{B}_{3}\\}:=\\{z_{1},z_{2},z_{1}+z_{2}-s\\}$
. The corresponding singular hypersurfaces $\mathcal{S}_{i}$ defined in eq.
(1) intersect at $3+2+1=6$ points in $\mathbb{C}\mathrm{P}^{2}$ (see Figure
2a):

$\displaystyle\mathbb{P}_{\omega}=\begin{cases}(z_{1},z_{2})\in\bigl{\\{}(0,0),(s,0),(0,s)\bigr{\\}}&\text{in
$U_{z}$}\ ,\\\ (x_{1},x_{2})\in\bigl{\\{}(0,0),(-1,0)\bigr{\\}}&\text{in
$U_{x}$}\ ,\\\ (y_{1},y_{2})=(0,0)&\text{in $U_{y}$}\ .\end{cases}$ (51)

In this example we consider intersection numbers between generic
$\varphi_{L}^{(\mathbf{2})}$ and $\varphi_{R}^{(\mathbf{2})}$ cocyles:

$\displaystyle\varphi^{(\mathbf{2})}_{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{1.0}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{1.0}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{1.0}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{1.0}{$\scriptscriptstyle\bullet$}}}}}}=z_{1}^{n_{1}}z_{2}^{n_{2}}\>(z_{1}+z_{2}-s)^{n_{3}}\>\mathrm{d}z_{1}\wedge\mathrm{d}z_{2}\
,$ (52)

where
$\mathchoice{\mathbin{\vbox{\hbox{\scalebox{1.0}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{1.0}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{1.0}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{1.0}{$\scriptscriptstyle\bullet$}}}}}\in\\{L,R\\}$
, and $n_{i}\in\mathbb{Z}$ .

Our task is to compute the multivariate residues shown in the main formula
(13) at each of the points (51). One way to do this is to find coordinate
transformations that localize on these points $\mathbb{P}_{\omega}$ , and
allow for direct series expansion and solution of the $n$PDE (14) either using
the algebraic formula (36) or the integral formula (22) (see also an
alternative algebraic formula in Appendix B). We collect such transformations
in Table 2 and refer the interested reader to Appendix C for further details.

Chart | Point | Coordinate transformation(s)
---|---|---
$U_{z}$ | $1:(0,0)$ | $(z_{1},\>z_{2})$
$2:(s,0)$ | $\bigl{\\{}(s+z_{1}z_{2},\>z_{2}),\>(s+z_{1},\>z_{1}z_{2}),\>(s+(z_{1}-1)z_{2},\>z_{2})\bigr{\\}}$
$3:(0,s)$ | $\bigl{\\{}(z_{1}z_{2},\>s+z_{2}),\>(z_{1},\>s+z_{1}z_{2}),\>((z_{1}-1)z_{2},\>s+z_{2})\bigr{\\}}$
$U_{x}$ | $4:(0,0)$ | $(\tfrac{z_{1}}{z_{2}},\>\tfrac{1}{z_{2}})$
$5:(-1,0)$ | $\bigl{\\{}(\tfrac{{z_{1}z_{2}-1}}{z_{2}},\>\tfrac{1}{z_{2}}),\>(\tfrac{{z_{1}-1}}{{z_{1}z_{2}}},\>\tfrac{1}{{z_{1}z_{2}}}),\>(\tfrac{{(s+z_{1})z_{2}}-1}{z_{2}},\>\tfrac{1}{z_{2}})\bigr{\\}}$
$U_{y}$ | $6:(0,0)$ | $(\tfrac{1}{z_{1}},\>\tfrac{z_{2}}{z_{1}})$
Table 2:  Intersection points $\mathbb{P}_{\omega}$ (middle) from the three
coordinate charts of $\mathbb{C}\mathrm{P}^{2}$ (left) that contribute to the
intersection numbers between forms (52) with the massless sunrise twist (50).
On the right we show the coordinate transformations used to compute the
residues (13). We sum over all the contributions of each displayed
transformation.

(a) Sunrise diagram

(b) Planar box diagram

(c) ${}_{3}F_{2}$

Figure 2:  Singular hypersurfaces $\mathcal{S}_{i}$ of the massless sunrise
twist (50) on the left, massless 2-loop planar box twist (64) in the middle,
and the ${}_{3}F_{2}$ twist (72) on the right. The red sphere (with identified
antipodal points) depicts the real slice of the projective plane
$\mathbb{C}\mathrm{P}^{2}$, whose equator is the line at infinity. The blue
dots represent the intersection points collected in eqs. (51, 66, 73).

######  Example 1.

Let us compute $\langle\varphi_{L}|\varphi_{R}\rangle$ with

$\displaystyle\hat{\varphi}_{L}=\frac{1}{z_{1}z_{2}}\,,\qquad\hat{\varphi}_{R}=\frac{1}{z_{1}z_{2}}\
.$ (53)

In the “right rescaling” scheme, the first step is to compute
$\Phi=\varphi_{L}\hat{\varphi}_{R}$ at each of the points in Table 2 and
identify the leading term corresponding to the
$\boldsymbol{\mu}_{\boldsymbol{\phi}}$-vector. The result is given in Table 3.

point | $\hat{\Phi}$ | ${-}\boldsymbol{\mu}_{\boldsymbol{\phi}}{-}\boldsymbol{2}$
---|---|---
1 | $z_{1}^{-2}z_{2}^{-2}$ | $\\{0,0\\}$
2a | $(s{+}z_{1}z_{2})^{-2}$ | $\\{-2,-2\\}$
2b | $(s{+}z_{1})^{-2}z_{2}^{-2}$ | $\\{-2,0\\}$
2c | $(s{+}z_{2}(z_{1}{-}1))^{-2}\\!$ | $\\{-2,-2\\}$
3a | $z_{1}^{-2}(s{+}z_{2})^{-2}$ | $\\{0,-2\\}$
3b | $(s{+}z_{1}z_{2})^{-2}$ | $\\{-2,-2\\}$

point $\hat{\Phi}$ ${-}\boldsymbol{\mu}_{\boldsymbol{\phi}}{-}\boldsymbol{2}$
3c $(z_{1}{-}1)^{-2}(z_{2}{+}s)^{-2}$ $\\{-2,-2\\}$ 4 $z_{1}^{-2}z_{2}^{-2}$
$\\{0,0\\}$ 5a $(z_{1}z_{2}{-}1)^{-2}$ $\\{-2,-2\\}$ 5b
$(z_{1}{-}1)^{-2}z_{2}^{-2}$ $\\{-2,0\\}$ 5c $((s{+}z_{1})z_{2}{-}1)^{-2}$
$\\{-2,-2\\}$ 6 $z_{1}^{-2}z_{2}^{-2}$ $\\{0,0\\}$

Table 3: Values for $\hat{\Phi}$ and $\boldsymbol{\mu}_{\boldsymbol{\phi}}$ at
the twelve sub-intersection points for Example 1.

We see from the general expression (36) that a given point will only
contribute if
$-\boldsymbol{\mu}_{\boldsymbol{\phi}}-\boldsymbol{2}\geq\boldsymbol{0}$. So
we realize that there only will be contributions from three of the
intersection points: 1, 4, and 6. In each case eq. (36) will contain one term
only, of the form

$\displaystyle\text{Res}(f)=\frac{\phi_{-2,-2}}{(v_{1}{-}1)(v_{2}{-}1)}\ .$
(54)

In each of the three points we have $\phi_{-2,-2}=1$, the only difference is
the value of the $v_{i}$ (that we remember have to be recomputed locally). We
get the expressions for $\text{Res}(f)$:

$\displaystyle\text{Point 1}:\frac{1}{\gamma_{1}\gamma_{2}}\qquad\text{Point
4}:\frac{-1}{\gamma_{1}(\gamma_{1}{+}\gamma_{2}{+}\gamma_{3})}\qquad\text{Point
6}:\frac{-1}{\gamma_{2}(\gamma_{1}{+}\gamma_{2}{+}\gamma_{3})}\ ,$ (55)

and adding them gives the intersection number

$\displaystyle\langle\varphi_{R}|\varphi_{L}\rangle$
$\displaystyle=\frac{\gamma_{3}}{\gamma_{1}\gamma_{2}(\gamma_{1}{+}\gamma_{2}{+}\gamma_{3})}\
,$ (56)

which is the correct result.

######  Example 2.

Consider now the intersection $\langle\varphi_{L}|\varphi_{R}\rangle$ between:

$\displaystyle\hat{\varphi}_{L}=\frac{1}{z_{1}^{2}z_{2}}\
,\qquad\hat{\varphi}_{R}=\frac{1}{z_{1}^{2}z_{2}}\ .$ (57)

Out of the six points collected in eq. (51) only 2 get non-trivial residues,
so we get:

$\displaystyle\langle\varphi_{L}|\varphi_{R}\rangle={\rm
Res}_{z=(0,0)}(\psi\varphi_{R})+{\rm
Res}_{z=(0,s)}(\psi\varphi_{R})=-\frac{1}{s^{2}}\frac{\gamma_{3}(\gamma_{1}+\gamma_{3})}{\gamma_{1}(1-\gamma_{1}^{2})\gamma_{2}}-\frac{1}{s^{2}}\frac{\gamma_{3}}{\gamma_{1}(1-\gamma_{1}^{2})}\
.$ (58)

######  Example 3.

We can also compute $\langle\varphi_{L}|\varphi_{R}\rangle$ of meromorphic
forms containing the $\mathcal{B}_{3}$ factor:

$\displaystyle\hat{\varphi}_{L}=\frac{1}{z_{1}z_{2}(z_{1}+z_{2}-s)}\
,\qquad\hat{\varphi}_{R}=\frac{1}{z_{1}z_{2}(z_{1}+z_{2}-s)^{2}}\ .$ (59)

Out of the six points shown in eq. (51) the 3 points from the $U_{z}$ chart
get non-zero residues:

$\displaystyle{\rm Res}_{z=(0,0)}(\psi\varphi_{R})$
$\displaystyle=-\tfrac{1}{s^{3}}\tfrac{1}{\gamma_{1}\gamma_{2}}\ ,$ (60)
$\displaystyle{\rm Res}_{z=(s,0)}(\psi\varphi_{R})$
$\displaystyle=0-\tfrac{1}{s^{3}}\tfrac{1+\gamma_{1}+2\gamma_{2}+2\gamma_{3}}{\gamma_{2}(\gamma_{2}+\gamma_{3})(1+\gamma_{2}+\gamma_{3})}-\tfrac{1}{s^{3}}\Bigl{(}\tfrac{1-\gamma_{1}}{(1+\gamma_{3})(1+\gamma_{2}+\gamma_{3})}+\tfrac{1+\gamma_{1}}{\gamma_{3}(\gamma_{2}+\gamma_{3})}\Bigr{)}\
,$ (61) $\displaystyle{\rm Res}_{z=(0,s)}(\psi\varphi_{R})$
$\displaystyle=-\tfrac{1}{s^{3}}\tfrac{1+2\gamma_{1}+\gamma_{2}+2\gamma_{3}}{\gamma_{1}(\gamma_{1}+\gamma_{3})(1+\gamma_{1}+\gamma_{3})}+0-\tfrac{1}{s^{3}}\tfrac{(1+\gamma_{1}+2\gamma_{3})(1+2\gamma_{1}+\gamma_{2}+2\gamma_{3})}{\gamma_{3}(1+\gamma_{3})(\gamma_{1}+\gamma_{3})(1+\gamma_{1}+\gamma_{3})}\
,$ (62)

where we separated contributions of the transformations shown in Table 2 into
individual terms. Summing up, the intersection number becomes:

$\displaystyle\langle\varphi_{L}|\varphi_{R}\rangle$ $\displaystyle={\rm
Res}_{z=(0,0)}(\psi\varphi_{R})+{\rm Res}_{z=(s,0)}(\psi\varphi_{R})+{\rm
Res}_{z=(0,s)}(\psi\varphi_{R})$
$\displaystyle=-\frac{1}{s^{3}}\frac{(\gamma_{1}+\gamma_{2}+\gamma_{3})(1+\gamma_{1}+\gamma_{2}+\gamma_{3})}{\gamma_{1}\gamma_{2}\gamma_{3}(1+\gamma_{3})}\
.$ (63)

###  5.3 Two-loop planar box diagram

Now we turn to the massless 2-loop planar box on the maximal cut in the Baikov
representation with the twist:

$\displaystyle
u=\begin{cases}z_{1}^{\gamma_{1}}z_{2}^{\gamma_{2}}\>(st+s(z_{1}+z_{2})+z_{1}z_{2})^{\gamma_{3}}&\text{in
$U_{z}$}\ ,\\\
x_{1}^{\gamma_{1}}x_{2}^{-\gamma_{0}-\gamma_{3}}\,\bigl{(}x_{1}+sx_{2}+sx_{1}x_{2}+stx_{2}^{2}\bigr{)}^{\gamma_{3}}&\text{in
$U_{x}$}\ ,\\\
y_{1}^{-\gamma_{0}-\gamma_{3}}y_{2}^{\gamma_{2}}\,\bigl{(}sy_{1}+y_{2}+sy_{1}y_{2}+sty_{1}^{2}\bigr{)}^{\gamma_{3}}&\text{in
$U_{y}$}\ ,\end{cases}$ (64)

where $\gamma_{0}:=\gamma_{1}+\gamma_{2}+\gamma_{3}$, and the factors (6) are:

$\displaystyle\\{\mathcal{B}_{1},\mathcal{B}_{2},\mathcal{B}_{3}\\}:=\\{z_{1},z_{2},st+s(z_{1}+z_{2})+z_{1}\\}\
.$ (65)

There are $3+1+1=5$ intersection points in $\mathbb{C}\mathrm{P}^{2}$ of the
singular hypersurfaces $\mathcal{S}_{i}$ (see Figure 2b), their location is as
follows:

$\displaystyle\mathbb{P}_{\omega}=\begin{cases}(z_{1},z_{2})\in\bigl{\\{}(0,0),(-t,0),(0,-t)\bigr{\\}}&\text{in
$U_{z}$}\ ,\\\ (x_{1},x_{2})=(0,0)&\text{in $U_{x}$}\ ,\\\
(y_{1},y_{2})=(0,0)&\text{in $U_{y}$}\ .\end{cases}$ (66)

Here we only consider intersection numbers between monomial
$\varphi_{L}^{(\mathbf{2})}$ and $\varphi_{R}^{(\mathbf{2})}$ cocycles:

$\displaystyle\varphi^{(\mathbf{2})}_{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{1.0}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{1.0}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{1.0}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{1.0}{$\scriptscriptstyle\bullet$}}}}}}=z_{1}^{n_{1}}z_{2}^{n_{2}}\>\mathrm{d}z_{1}\wedge\mathrm{d}z_{2}\
,$ (67)

with the same notation as in eq. (52).

Some of the intersection points (66) turn out to be degenerate: they have
three singular hypersurfaces $\mathcal{S}_{i}$ passing through them. An
example of this is the point $x=(0,0)$ located on the line at infinity, as can
be seen from eq. (64) and Figure 2b. One way to amend this issue is to employ
the resolution of singularities procedure (closely related to the sector
decomposition algorithm, see Borowka:2017idc ; Heinrich:2021dbf ;
Bogner:2007cr ). In Appendix C we give further details and collect the full
list of coordinate transformations used for computation of intersection
numbers in this setup.

######  Example 4.

As in Example 1, let us consider the intersection number
$\langle\varphi_{L}|\varphi_{R}\rangle$ between:

$\displaystyle\hat{\varphi}_{L}=\frac{1}{z_{1}z_{2}}\
,\qquad\hat{\varphi}_{R}=\frac{1}{z_{1}z_{2}}\ .$ (68)

Only the origins of the three charts (66) contribute to this intersection
number producing:

$\displaystyle\langle\varphi_{L}|\varphi_{R}\rangle$ $\displaystyle={\rm
Res}_{z=(0,0)}(\psi\varphi_{R})+{\rm Res}_{x=(0,0)}(\psi\varphi_{R})+{\rm
Res}_{y=(0,0)}(\psi\varphi_{R})$
$\displaystyle=\tfrac{1}{\gamma_{1}\gamma_{2}}+\Bigl{(}\tfrac{-1}{\gamma_{1}(\gamma_{2}+\gamma_{3})}+\tfrac{1}{(\gamma_{2}+\gamma_{3})(\gamma_{0}+\gamma_{3})}\Bigr{)}+\Bigl{(}\tfrac{1}{(\gamma_{1}+\gamma_{3})(\gamma_{0}+\gamma_{3})}+\tfrac{-1}{\gamma_{2}(\gamma_{1}+\gamma_{3})}\Bigr{)}\
.$ (69)

######  Example 5.

Similar to Example 2, we compute the intersection between:

$\displaystyle\hat{\varphi}_{L}=\frac{1}{z_{1}^{2}z_{2}}\
,\qquad\hat{\varphi}_{R}=\frac{1}{z_{1}^{2}z_{2}}\ .$ (70)

We find the three points from $\mathbb{P}_{\omega}$ contribute to this
intersection number, resulting in:

$\displaystyle\langle\varphi_{L}|\varphi_{R}\rangle$ $\displaystyle={\rm
Res}_{z=(0,0)}(\psi\varphi_{R})+{\rm Res}_{z=(0,-t)}(\psi\varphi_{R})+{\rm
Res}_{x=(0,0)}(\psi\varphi_{R})$
$\displaystyle=-\tfrac{1}{t^{2}}\tfrac{\gamma_{3}(\gamma_{1}+\gamma_{3})}{\gamma_{1}\gamma_{2}(1-\gamma_{1}^{2})}-\tfrac{1}{s^{2}t^{2}}\tfrac{\gamma_{3}(s-t)^{2}}{\gamma_{1}(1-\gamma_{1}^{2})}+\tfrac{1}{s^{2}}\tfrac{\gamma_{3}(\gamma_{1}+\gamma_{3})}{\gamma_{1}(1-\gamma_{1}^{2})(\gamma_{2}+\gamma_{3})}\
.$ (71)

Chart | Point | Coordinate transformation(s)
---|---|---
$U_{z}$ | $(0,0)$ | $\bigl{\\{}(z_{1}z_{2},\>z_{2}),(z_{1},\>z_{1}z_{2}),((1+z_{1})z_{2},\>z_{2})\bigr{\\}}$
$(0,s)$ | $(z_{1},\>s+z_{2})$
$(s,s)$ | $\bigl{\\{}(s+z_{1}z_{2},\>s+z_{2}),(s+z_{1},\>s+z_{1}z_{2}),(s+(1+z_{1})z_{2},\>s+z_{2})\bigr{\\}}$
$(1,0)$ | $(1+z_{1},\>z_{2})$
$(1,s)$ | $(1+z_{1},\>s+z_{2})$
$(1,1)$ | $\bigl{\\{}(1+z_{1}z_{2},\>1+z_{2}),(1+z_{1},\>1+z_{1}z_{2}),(1+(1+z_{1})z_{2},\>1+z_{2})\bigr{\\}}$
$U_{x}$ | $(0,0)$ | $\bigl{\\{}(z_{1},\>\tfrac{1}{z_{2}}),(\tfrac{1}{z_{2}},\>\tfrac{1}{z_{1}z_{2}}),(1+z_{1},\>\tfrac{1}{z_{2}})\bigr{\\}}$
$(1,0)$ | $(\tfrac{1+z_{1}}{z_{2}},\>\tfrac{1}{z_{2}})$
$U_{y}$ | $(0,0)$ | $\bigl{\\{}(\tfrac{1}{z_{1}z_{2}},\>\tfrac{1}{z_{1}}),(\tfrac{1}{z_{1}},\>z_{2}),(\tfrac{1}{(s^{-1}+z_{1})z_{2}},\tfrac{1}{s^{-1}+z_{1}})\bigr{\\}}$
Table 4:  Intersection points $\mathbb{P}_{\omega}$ (middle) from the three
coordinate charts of $\mathbb{C}\mathrm{P}^{2}$ (left) that contribute to the
intersection numbers between forms (74) with the ${}_{3}F_{2}$ twist (72). On
the right we show the coordinate transformations used to compute the residues
(13). We sum over all the contributions of each displayed transformation.

###  5.4 ${}_{3}F_{2}$ hypergeometric function

Another setup with degenerate intersection points is the ${}_{3}F_{2}$
hypergeometric function, whose twist looks like this:

$\displaystyle
u=\begin{cases}z_{1}^{\gamma_{1}}z_{2}^{\gamma_{2}}\>(1-z_{1})^{\gamma_{3}}\>(s-z_{2})^{\gamma_{4}}\>(z_{1}-z_{2})^{\gamma_{5}}&\text{in
$U_{z}$}\ ,\\\
(-1)^{\gamma_{3}+\gamma_{5}}\>x_{1}^{\gamma_{1}}x_{2}^{-\gamma_{0}}\>(x_{1}-x_{2})^{\gamma_{3}}\>(sx_{2}-1)^{\gamma_{4}}\>(1-x_{1})^{\gamma_{5}}&\text{in
$U_{x}$}\ ,\\\
(-1)^{\gamma_{3}}\>y_{1}^{-\gamma_{0}}y_{2}^{\gamma_{2}}\>(1-y_{1})^{\gamma_{3}}\>(sy_{1}-y_{2})^{\gamma_{4}}\>(1-y_{2})^{\gamma_{5}}&\text{in
$U_{y}$}\ ,\end{cases}$ (72)

where $\gamma_{0}:=\gamma_{1}+\ldots+\gamma_{5}$ . The five singular
hypersurfaces $\mathcal{S}_{i}$ intersect at the following $6+2+1=9$ points
(see Figure 2c):

$\displaystyle\mathbb{P}_{\omega}=\begin{cases}(z_{1},z_{2})\in\bigl{\\{}(0,0),(0,s),(s,s),(1,0),(1,s),(1,1)\bigr{\\}}&\text{in
$U_{z}$}\ ,\\\ (x_{1},x_{2})\in\bigl{\\{}(0,0),(1,0)\bigr{\\}}&\text{in
$U_{x}$}\ ,\\\ (y_{1},y_{2})=(0,0)&\text{in $U_{y}$}\ .\end{cases}$ (73)

We compute intersection numbers between generic $\varphi_{L}^{(\mathbf{2})}$
and $\varphi_{R}^{(\mathbf{2})}$ cocyles:

$\displaystyle\varphi^{(\mathbf{2})}_{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{1.0}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{1.0}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{1.0}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{1.0}{$\scriptscriptstyle\bullet$}}}}}}=z_{1}^{n_{1}}z_{2}^{n_{2}}\>(1-z_{1})^{n_{3}}(s-z_{2})^{n_{4}}(z_{1}-z_{2})^{n_{5}}\>\mathrm{d}z_{1}\wedge\mathrm{d}z_{2}\
,$ (74)

with the same notation as in eq. (52).

######  Example 6.

In continuation of Example 1 and Example 4, we calculate the intersection of:

$\displaystyle\hat{\varphi}_{L}=\frac{1}{z_{1}z_{2}}\
,\qquad\hat{\varphi}_{R}=\frac{1}{z_{1}z_{2}}\ .$ (75)

Once again, only the origins of the charts contribute to this intersection
number giving the total:

$\displaystyle\langle\varphi_{L}|\varphi_{R}\rangle$ $\displaystyle={\rm
Res}_{z=(0,0)}(\psi\varphi_{R})+{\rm Res}_{x=(0,0)}(\psi\varphi_{R})+{\rm
Res}_{y=(0,0)}(\psi\varphi_{R})$
$\displaystyle=\Bigl{(}\tfrac{1}{\gamma_{1}(\gamma_{1}+\gamma_{2}+\gamma_{5})}+\tfrac{1}{\gamma_{2}(\gamma_{1}+\gamma_{2}+\gamma_{5})}+0\Bigr{)}$
(76)
$\displaystyle\quad+\Bigl{(}\tfrac{-1}{\gamma_{1}(\gamma_{2}+\gamma_{4}+\gamma_{5})}+\tfrac{1}{\gamma_{0}(\gamma_{2}+\gamma_{4}+\gamma_{5})}+0\Bigr{)}+\Bigl{(}\tfrac{1}{\gamma_{0}(\gamma_{1}+\gamma_{3}+\gamma_{5})}+\tfrac{-1}{\gamma_{2}(\gamma_{1}+\gamma_{3}+\gamma_{5})}+0\Bigr{)}\
.$

This concludes our collection of examples, for intersection numbers in $n=2$
variables, involving Feynman integrals and other mathematical functions. Let
us remark, that the presented method is also applicable to meromorphic
$n$-forms in higher dimensions.

##  6 Conclusion

In this work we proposed a new computational method for the evaluation of
intersection numbers for twisted meromorphic $n$-forms through Stokes’
theorem, which is based on the solution of a $n$-th order partial differential
equation ($n$PDE). Our finding can be summarised simply as:

$\langle\varphi_{L}|\varphi_{R}\rangle=\frac{1}{(2\pi
i)^{n}}\int(u\,\varphi_{L})\wedge(u^{-1}\,\varphi_{R})=\frac{1}{(2\pi
i)^{n}}\oint(u^{-1}\,\varphi_{R})\int(u\,\varphi_{L})\ .$ (77)

The evaluation of the last integral is performed by multivariate residues, by
using multivariate Laurent series expansions. The analytic properties of the
intersection numbers and of the $n$PDE yielded the algebraic determination of
the contributing residues, which we were able to cast in closed forms for an
efficient evaluation.

The presented method requires the knowledge of the intersection points as
input. In the case of a twist corresponding to a hyper-plane arrangement and
normal crossing at the intersection points, this information can be easily
extracted from the integrand. Instead, for generic configurations, when more
hypersurfaces pass through a given intersection point, the desingularization
procedure constitutes a challenging, hence interesting, mathematical problem.

The formulation of the intersection number for twisted cohomology presented
here applies to the case of regulated singularities. We are confident that it
can be extended to the relative cohomolgy case, where the singularities of the
integral are not regulated within the twist.

The new method presented here can be applied to derive linear relations,
differential equations, difference equations, as well as quadratic relations
for Feynman integrals, as well as for wider class of functions, such as
Aomoto-Gel’fand integrals, Euler-Mellin integrals, GKZ hypergeometric
functions, and related functions, therefore can be of interest in
computational (quantum) field theory and in computational differential and
algebraic topology.

### Acknowledgements

We wish to acknowledge interesting and stimulating discussions on intersection
theory with Sergio Cacciatori, Yoshiaki Goto, Saiei Matsubara-Heo, Keiji
Matsumoto, and Nobuki Takayama, at various stages. We would like to thank
Sebastian Mizera for discussions and comments on the manuscript, and Henrik
Munch for many good discussions and various checks. H.F. would like to thank
Cristian Vergu for stimulating discussions.
V.C. is supported by the Diagrammalgebra Stars Wild-Card Grant UNIPD. H.F. is
partially supported by a Carlsberg Foundation Reintegration Fellowship, and
has received funding from the European Union’s Horizon 2020 research and
innovation program under the Marie Skłodowska-Curie grant agreement No. 847523
‘INTERACTIONS’. The work of M.K.M. is supported by Fellini - Fellowship for
Innovation at INFN funded by the European Union’s Horizon 2020 research and
innovation programme under the Marie Skłodowska-Curie grant agreement No
754496. F.G. is supported by Fondazione Cassa di Risparmio di Padova e Rovigo
(CARIPARO).

##  Appendix A Algebraic solution: proof

The purpose of this appendix is to show the validity of the algebraic solution
given by eq. (36) and discuss some features not mentioned in the main text.

###  A.1 Deriving the algebraic expression

We will consider a more general case than in Section 3, and only at the end
specify to the “right rescaling” framework. Our starting point will be eqs.
(13) and (21):

$\displaystyle\langle\varphi_{L}|\varphi_{R}\rangle=\sum_{p\in\mathbb{P}_{\omega}}{\rm
Res}_{z=p}(\psi\varphi_{R})$ (78)

where $\psi$ is defined through

$\displaystyle\hat{\nabla}_{\omega_{1}}\cdots\hat{\nabla}_{\omega_{n}}\psi=\hat{\varphi}_{L}\qquad\quad\text{where}\qquad\quad\hat{\nabla}_{\omega_{i}}:=\partial_{z_{i}}+(\partial_{z_{i}}\log(u))$
(79)

We want to approach the differential equation locally as a series expansions.
So in the following we will consider a point corresponding to one of the terms
of eq. (78) with the assumption that the proper variable changes has been made
so $p=0$ and in addition that the series expansions are well defined so no
further blowups are needed as also discussed in Section 3.

Let us write the objects involved as such expansions:

$\displaystyle\hat{\varphi}_{L}=\sum_{\boldsymbol{i}=\boldsymbol{\mu}_{\boldsymbol{L}}}^{\boldsymbol{\infty}}\\!\hat{\varphi}_{L,\boldsymbol{i}}\,z^{\boldsymbol{i}}\qquad\qquad\hat{\varphi}_{R}=\sum_{\boldsymbol{i}=\boldsymbol{\mu}_{\boldsymbol{R}}}^{\boldsymbol{\infty}}\\!\hat{\varphi}_{R,\boldsymbol{i}}z^{\boldsymbol{i}}\qquad\qquad\psi=\sum_{\boldsymbol{i}=\boldsymbol{\mu}_{\boldsymbol{\psi}}}^{\boldsymbol{\infty}}\\!\psi_{\boldsymbol{i}}z^{\boldsymbol{i}}$
(80) $\displaystyle
w:=\log(u)=\sum_{\boldsymbol{i}=\boldsymbol{\mu}_{\boldsymbol{w}}}^{\boldsymbol{\infty}}\\!w_{\boldsymbol{i}}z^{\boldsymbol{i}}\;+\;\sum_{j=1}^{n}v_{j}\log(z_{j})$
(81)

The exponent of the leading powers $\boldsymbol{\mu}_{\boldsymbol{L}}$,
$\boldsymbol{\mu}_{\boldsymbol{R}}$,
$\boldsymbol{\mu}_{\boldsymbol{w}}$777$\boldsymbol{\mu}_{\boldsymbol{w}}$ will
be $\boldsymbol{0}$ for all problems where $u=\prod\mathcal{B}^{\gamma}$ as
for the Feynman integrals in Baikov representation discussed in this paper.
Yet we keep it general in this appendix since the theory is valid also for
other functional forms for $u$. For instance a $u$ containing a factor of
$e^{1/z}$ will give $\mu_{w}=-1$. are given by the physical problem, while
$\boldsymbol{\mu}_{\boldsymbol{\psi}}$ is not since the $\psi$-expression is
an ansatz so $\boldsymbol{\mu}_{\boldsymbol{\psi}}$ is just assumed to chosen
to be small enough that all manipulations done in the following are valid.

With these expansions we may easily write the residue we are looking for as

$\displaystyle{\rm
Res}_{z=0}(\psi\varphi_{R})=\sum_{\boldsymbol{i}=\boldsymbol{\mu}_{\boldsymbol{\psi}}\\!\\!}^{-\boldsymbol{\mu}_{\boldsymbol{R}}-\boldsymbol{1}\,}\\!\\!\\!\psi_{\boldsymbol{i}}\,\hat{\varphi}_{R,-\boldsymbol{i}-\boldsymbol{1}}$
(82)

The challenging part is solving eq. (79) to identify the expressions for the
$\psi_{\boldsymbol{i}}$.

####  A.1.1 The case $n=2$.

Solving the nPDE. We first discuss the case $n=2$, and later we extend it to
arbitrary $n$-forms. Solving eq. (79) corresponds to solving $\Delta=0$ with

$\displaystyle\Delta:=\hat{\nabla}_{\omega_{1}}\hat{\nabla}_{\omega_{2}}\psi-\hat{\varphi}_{L}$
(83)

This corresponds to

$\displaystyle\Delta$
$\displaystyle=(\partial_{1}\partial_{2}\psi)+(\partial_{1}\partial_{2}w)\psi+(\partial_{1}w)(\partial_{2}\psi)+(\partial_{2}w)(\partial_{1}\psi)+(\partial_{1}w)(\partial_{2}w)\psi-\hat{\varphi}_{L}$
(84)

where we have used the obvious notation $\partial_{i}:=\partial_{z_{i}}$.
Inserting the expansions we get

$\displaystyle\Delta$
$\displaystyle=\;\,\sum_{\boldsymbol{i}=\boldsymbol{\mu}_{\boldsymbol{\psi}}}^{\boldsymbol{\infty}}(i_{1}i_{2}\psi_{\boldsymbol{i}}-\hat{\varphi}_{L,\boldsymbol{i}-\boldsymbol{1}})z_{1}^{i_{1}-1}z_{2}^{i_{2}-1}$
$\displaystyle\;\;+\sum_{\boldsymbol{i}=\boldsymbol{\mu}_{\boldsymbol{\psi}}}^{\boldsymbol{\infty}}\sum_{\boldsymbol{j}=\boldsymbol{\mu}_{\boldsymbol{w}}}^{\boldsymbol{\infty}}\big{(}j_{1}j_{2}w_{\boldsymbol{j}}+i_{2}(j_{1}w_{\boldsymbol{j}}+\delta_{\boldsymbol{j},\boldsymbol{0}}v_{1})+i_{1}(j_{2}w_{\boldsymbol{j}}+\delta_{\boldsymbol{j},\boldsymbol{0}}v_{2})\big{)}\psi_{\boldsymbol{i}}z_{1}^{i_{1}+j_{1}-1}z_{2}^{i_{2}+j_{2}-1}$
$\displaystyle\;\;+\sum_{\boldsymbol{i}=\boldsymbol{\mu}_{\boldsymbol{\psi}}}^{\boldsymbol{\infty}}\sum_{\boldsymbol{j}=\boldsymbol{\mu}_{\boldsymbol{w}}}^{\boldsymbol{\infty}}\sum_{\boldsymbol{k}=\boldsymbol{\mu}_{\boldsymbol{w}}}^{\boldsymbol{\infty}}\\!(j_{1}w_{\boldsymbol{j}}+\delta_{\boldsymbol{j},\boldsymbol{0}}v_{1})(k_{2}w_{\boldsymbol{k}}+\delta_{\boldsymbol{k},\boldsymbol{0}}v_{2})\psi_{\boldsymbol{i}}\,z_{1}^{i_{1}+j_{1}+k_{1}-1}z_{2}^{i_{2}+j_{2}+k_{2}-1}$
(85)

Changing the summation variables we may rewrite this as

$\displaystyle\Delta$
$\displaystyle=\\!\\!\\!\sum_{\boldsymbol{s}=\boldsymbol{\mu}_{\boldsymbol{\psi}}+n\boldsymbol{\mu}_{\boldsymbol{w}}\\!}\\!\\!\\!\Delta_{\boldsymbol{s}}\,z^{\boldsymbol{s}-\boldsymbol{1}}$
(86)

where888In eqs. (86) and (87) we have reinserted $n$ reflecting the way the
corresponding equations will look in the $n$-variate case.

$\displaystyle\Delta_{\boldsymbol{s}}$
$\displaystyle:=\sum_{\boldsymbol{t}=\boldsymbol{\mu}_{\boldsymbol{\psi}}}^{\boldsymbol{s}-n\boldsymbol{\mu}_{\boldsymbol{w}}}\\!\\!R(\boldsymbol{s}{-}\boldsymbol{t},\boldsymbol{t})\,\psi_{\boldsymbol{t}}\;\;-\;\;\hat{\varphi}_{L,\boldsymbol{s}-\boldsymbol{1}}$
(87)

with

$\displaystyle R(\boldsymbol{\alpha},\boldsymbol{\beta})$
$\displaystyle=(\beta_{1}{+}v_{1})(\beta_{2}{+}v_{2})\delta_{\boldsymbol{\alpha},\boldsymbol{0}}+\big{(}\alpha_{1}\alpha_{2}+\alpha_{1}(\beta_{2}{+}v_{2})+\alpha_{2}(\beta_{1}{+}v_{1})\big{)}w_{\boldsymbol{\alpha}}$
$\displaystyle\;\;\;\,+\sum_{\boldsymbol{j}=\boldsymbol{\mu}_{\boldsymbol{w}}}^{\boldsymbol{\alpha}-\boldsymbol{\mu}_{\boldsymbol{w}}}(\alpha_{1}{-}j_{1})j_{2}\,w_{\boldsymbol{\alpha}-\boldsymbol{j}}w_{\boldsymbol{j}}$
(88)

which reduces to the expression given in eq. (39) in the case where
$\boldsymbol{\mu}_{\boldsymbol{w}}=\boldsymbol{0}$. In general this function
$R$ may be interpreted as $R(\boldsymbol{\alpha},\boldsymbol{\beta})$ being
the coefficient in $\Delta$ of
$\psi_{\boldsymbol{\beta}}\,z^{\boldsymbol{\alpha}{+}\boldsymbol{\beta}{-}\boldsymbol{1}}$.

Having $\Delta=0$ in general requires that each term in the $z$-expansion of
eq. (86) is zero independently. This means that our task now is to solve
$\Delta_{\boldsymbol{s}}=0$ for each value of $\boldsymbol{s}$.

This can be done. If we define

$\displaystyle\tilde{\boldsymbol{\mu}}_{\boldsymbol{\psi}}:=\boldsymbol{\mu}_{\boldsymbol{L}}-n\boldsymbol{\mu}_{\boldsymbol{w}}+\boldsymbol{1}$
(89)

we find the recursive solution

$\displaystyle\psi_{\boldsymbol{q}}\;$
$\displaystyle=\;\qquad\qquad\qquad\qquad\quad
0\qquad\qquad\qquad\qquad\qquad\qquad\text{if}\;\;\boldsymbol{q}\ngeq\tilde{\boldsymbol{\mu}}_{\boldsymbol{\psi}}$
$\displaystyle\psi_{\boldsymbol{q}}\;$
$\displaystyle=\;\bigg{(}\hat{\varphi}_{L,\boldsymbol{q}+n\boldsymbol{\mu}_{\boldsymbol{w}}-\boldsymbol{1}}\;-\\!\\!\\!\\!\\!\sum_{\boldsymbol{r}:\;\tilde{\boldsymbol{\mu}}_{\boldsymbol{\psi}}\leq\boldsymbol{r}<\boldsymbol{q}}\\!\\!\\!\\!\\!Q(\boldsymbol{q},\boldsymbol{r})\,\psi_{\boldsymbol{r}}\bigg{)}\Big{/}Q(\boldsymbol{q},\boldsymbol{q})\qquad\text{if}\;\;\boldsymbol{q}\geq\tilde{\boldsymbol{\mu}}_{\boldsymbol{\psi}}$
(90)

where

$\displaystyle
Q(\boldsymbol{q},\boldsymbol{r}):=R(\boldsymbol{q}{-}\boldsymbol{r}{+}n\boldsymbol{\mu}_{\boldsymbol{w}}\,,\;\boldsymbol{r})$
(91)

and where the exact definitions of $\geq$ and other binary operators applied
to the index vectors are given is Section A.4. The quantity
$\tilde{\boldsymbol{\mu}}_{\boldsymbol{\psi}}$ may be interpreted as the
“correct” value for $\boldsymbol{\mu}_{\boldsymbol{\psi}}$, or at least as a
value for which
$\boldsymbol{\mu}_{\boldsymbol{\psi}}\nleq\tilde{\boldsymbol{\mu}}_{\boldsymbol{\psi}}$
would make it impossible for the above derivation to go through.

The recursive solution given by eq. (90) can be useful on its own to find
$\psi$-solutions to use as arguments of the residue function using for
instance eq. (82), but let us continue the derivation in order to find a
closed expression.

Evaluation of the residue. The next step is to re-express eq. (90) in a form
in which only the coefficients of the individual
$\hat{\varphi}_{L,\boldsymbol{i}}$ are recursive. This can be done as

$\displaystyle\psi_{\boldsymbol{q}}$
$\displaystyle=\sum_{\boldsymbol{\kappa}=\tilde{\boldsymbol{\mu}}_{\boldsymbol{\psi}}}^{\boldsymbol{q}}\\!\\!\hat{\varphi}_{L,\boldsymbol{\kappa}{+}n\boldsymbol{\mu}_{\boldsymbol{w}}-\boldsymbol{1}}\,\Psi(\boldsymbol{q},\boldsymbol{q}{-}\boldsymbol{\kappa})$
(92)

with

$\displaystyle\Psi(\boldsymbol{q},\boldsymbol{\tau})\;=\left\\{\begin{array}[]{cc}0&\;\;\text{if}\;\;\boldsymbol{\tau}\ngeq\boldsymbol{0}\\\
1/Q(\boldsymbol{q},\boldsymbol{q})&\;\;\text{if}\;\;\boldsymbol{\tau}=\boldsymbol{0}\\\
\frac{-\sum_{\boldsymbol{i}:\,\boldsymbol{0}\leq\boldsymbol{i}<\boldsymbol{\tau}}Q(\boldsymbol{q}{-}\boldsymbol{i},\boldsymbol{q}{-}\boldsymbol{\tau})\Psi(\boldsymbol{q},\boldsymbol{i})}{Q(\boldsymbol{q}{-}\boldsymbol{\tau},\boldsymbol{q}{-}\boldsymbol{\tau})}&\;\;\text{if}\;\;\boldsymbol{\tau}>\boldsymbol{0}\end{array}\right.$
(96)

or alternatively

$\displaystyle\Psi(\boldsymbol{q},\boldsymbol{\tau})\;=\left\\{\begin{array}[]{cc}0&\;\;\text{if}\;\;\boldsymbol{\tau}\ngeq\boldsymbol{0}\\\
1/W(\boldsymbol{q},\boldsymbol{0})&\;\;\text{if}\;\;\boldsymbol{\tau}=\boldsymbol{0}\\\
\frac{-\sum_{\boldsymbol{i}:\,\boldsymbol{0}\leq\boldsymbol{i}<\boldsymbol{\tau}}W(\boldsymbol{q}{-}\boldsymbol{\tau},\boldsymbol{\tau}{-}\boldsymbol{i})\Psi(\boldsymbol{q},\boldsymbol{i})}{W(\boldsymbol{q}{-}\boldsymbol{\tau},\boldsymbol{0})}&\;\;\text{if}\;\;\boldsymbol{\tau}>\boldsymbol{0}\end{array}\right.$
(100)

where
$W(\boldsymbol{\eta},\boldsymbol{\lambda}):=Q(\boldsymbol{\eta}{+}\boldsymbol{\lambda},\boldsymbol{\eta})$
or correspondingly

$\displaystyle W(\boldsymbol{\eta},\boldsymbol{\lambda})$
$\displaystyle=R(\boldsymbol{\lambda}{+}n\boldsymbol{\mu}_{\boldsymbol{w}},\,\boldsymbol{\eta})$
(101)

Eq. (100) is a recursive expression that only involves the coefficients of the
expansion of $w$. The recursion can be solved with the result

$\displaystyle\Psi(\boldsymbol{q},\boldsymbol{\tau})$
$\displaystyle=\sum_{\sigma\in\text{VC}(\boldsymbol{\tau})}\frac{(-1)^{|\sigma|}}{W(\boldsymbol{q},\boldsymbol{0})}\prod_{i=1}^{|\sigma|}\frac{W(\boldsymbol{q}{-}\boldsymbol{\tau}{+}\sum_{j<i}\boldsymbol{\sigma}_{\boldsymbol{j}},\,\boldsymbol{\sigma}_{\boldsymbol{i}})}{W(\boldsymbol{q}{-}\boldsymbol{\tau}{+}\sum_{j<i}\boldsymbol{\sigma}_{\boldsymbol{j}},\,\boldsymbol{0})}$
(102)

where VC refers to the vector compositions defined and discussed in Section
3.2. By combining eqs. (92) and (82), and changing the summation variables we
get

$\displaystyle\text{Res}(\psi\varphi_{R})$
$\displaystyle=\sum_{\boldsymbol{\tau}=\boldsymbol{0}}^{n\boldsymbol{\mu}_{\boldsymbol{w}}-\boldsymbol{\mu}_{\boldsymbol{L}}-\boldsymbol{\mu}_{\boldsymbol{R}}-\boldsymbol{2}\;}\;\;\sum_{\boldsymbol{h}=\tilde{\boldsymbol{\mu}}_{\boldsymbol{\psi}}}^{-\boldsymbol{\mu}_{\boldsymbol{R}}-\boldsymbol{\tau}-\boldsymbol{1}}\\!\\!\\!\;\Psi(\boldsymbol{h}{+}\boldsymbol{\tau},\boldsymbol{\tau})\;\hat{\varphi}_{L,\boldsymbol{h}+n\boldsymbol{\mu}_{\boldsymbol{w}}-\boldsymbol{1}}\;\hat{\varphi}_{R,-\boldsymbol{h}-\boldsymbol{\tau}-\boldsymbol{1}}$
(103)

and finally we may insert the expression for $\Psi$ from eq. (102) and we get
the most general form of the algebraic expression:

$\displaystyle\boxed{\text{Res}(\psi\varphi_{R})\;=\\!\\!\\!\sum_{\boldsymbol{\tau}=\boldsymbol{0}}^{n\boldsymbol{\mu}_{\boldsymbol{w}}-\boldsymbol{\mu}_{\boldsymbol{L}}-\boldsymbol{\mu}_{\boldsymbol{R}}-\boldsymbol{2}\;}\\!\\!\\!\sum_{\sigma\in\text{VC}(\boldsymbol{\tau})}\;\;\sum_{\boldsymbol{h}=\boldsymbol{\mu}_{\boldsymbol{L}}-n\boldsymbol{\mu}_{\boldsymbol{w}}+\boldsymbol{1}}^{-\boldsymbol{\mu}_{\boldsymbol{R}}-\boldsymbol{\tau}-\boldsymbol{1}}\\!\\!\\!\\!\\!Y(\boldsymbol{h},\boldsymbol{\tau},\sigma)\;\hat{\varphi}_{L,\boldsymbol{h}+n\boldsymbol{\mu}_{\boldsymbol{w}}-\boldsymbol{1}}\;\hat{\varphi}_{R,-\boldsymbol{h}-\boldsymbol{\tau}-\boldsymbol{1}}}$
(104)

where

$\displaystyle Y(\boldsymbol{h},\boldsymbol{\tau},\sigma)$
$\displaystyle:=\frac{(-1)^{|\sigma|}}{W(\boldsymbol{h}{+}\boldsymbol{\tau},\boldsymbol{0})}\prod_{i=1}^{|\sigma|}\frac{W(\boldsymbol{h}{+}\sum_{j<i}\boldsymbol{\sigma}_{\boldsymbol{j}},\,\boldsymbol{\sigma}_{\boldsymbol{i}})}{W(\boldsymbol{h}{+}\sum_{j<i}\boldsymbol{\sigma}_{\boldsymbol{j}},\,\boldsymbol{0})}$
(105)

We notice that $\boldsymbol{\tau}$ is not really needed as an argument to $Y$
since $\boldsymbol{\tau}=\sum_{j}\boldsymbol{\sigma}_{\boldsymbol{j}}$.

####  A.1.2 Beyond $n=2$.

Even though the derivation in the previous section was done in the case of
$n=2$, it will go through in the same way for different values. In particular
the final result eq. (104) will look the same for all values of $n$. The main
exceptions are eqs. (83, 84, 85, 88). Looking briefly at some other cases, for
$n=1$ we have

$\displaystyle\Delta\;=\;\hat{\nabla}_{\omega}\psi-\hat{\varphi}_{L}\;=\;(\partial_{z}\psi)+(\partial_{z}w)\psi-\hat{\varphi}_{L}$
(106)

which will correspond to

$\displaystyle R(\alpha,\beta)=(\beta{+}v)\delta_{\alpha,0}+\alpha w_{\alpha}$
(107)

as in eq. (38). Likewise for $n=3$ we have

$\displaystyle\Delta$
$\displaystyle=\hat{\nabla}_{\omega_{1}}\hat{\nabla}_{\omega_{2}}\hat{\nabla}_{\omega_{3}}\psi-\hat{\varphi}_{L}$
(108)
$\displaystyle=(\partial_{1}w)(\partial_{23}\psi)+(\partial_{2}w)(\partial_{13}\psi)+(\partial_{3}w)(\partial_{12}\psi)+(\partial_{12}w)(\partial_{3}\psi)+(\partial_{13}w)(\partial_{2}\psi)+(\partial_{23}w)(\partial_{1}\psi)$
$\displaystyle+\big{(}(\partial_{1}w)(\partial_{23}w)+(\partial_{2}w)(\partial_{13}w)+(\partial_{3}w)(\partial_{12}w)\big{)}\psi+(\partial_{1}w)(\partial_{2}w)(\partial_{3}\psi)+(\partial_{1}w)(\partial_{3}w)(\partial_{2}\psi)$
$\displaystyle+(\partial_{2}w)(\partial_{3}w)(\partial_{1}\psi)+(\partial_{123}w)\psi+(\partial_{1}w)(\partial_{2}w)(\partial_{3}w)\psi+\partial_{123}\psi-\hat{\varphi}_{L}$
(109)

which after a similar derivation yields

$\displaystyle R(\boldsymbol{\alpha},\boldsymbol{\beta})$
$\displaystyle=\;(\beta_{1}{+}v_{1})(\beta_{2}{+}v_{2})(\beta_{3}{+}v_{3})\delta_{\boldsymbol{\alpha},\boldsymbol{0}}$
$\displaystyle\;\;\;+\big{(}(\alpha_{1}{+}\beta_{1}{+}v_{1})(\alpha_{2}{+}\beta_{2}{+}v_{2})(\alpha_{3}{+}\beta_{3}{+}v_{3})-(\beta_{1}{+}v_{1})(\beta_{2}{+}v_{2})(\beta_{3}{+}v_{3})\big{)}w_{\boldsymbol{\alpha}}$
$\displaystyle\;\;\;+\;\sum_{\boldsymbol{j}=\boldsymbol{\mu}_{\boldsymbol{w}}}^{\boldsymbol{\alpha}{-}\boldsymbol{\mu}_{\boldsymbol{w}}}\big{(}(\alpha_{1}{-}j_{1})j_{2}(j_{3}{+}\beta_{3}{+}v_{3})+(\alpha_{2}{-}j_{2})j_{3}(j_{1}{+}\beta_{1}{+}v_{1})+(\alpha_{3}{-}j_{3})j_{1}(j_{2}{+}\beta_{2}{+}v_{2})\big{)}w_{\boldsymbol{\alpha}{-}\boldsymbol{j}}w_{\boldsymbol{j}}$
$\displaystyle\;\;\;+\,\sum_{\boldsymbol{j}=\boldsymbol{\mu}_{\boldsymbol{w}}}^{\boldsymbol{\alpha}-2\boldsymbol{\mu}_{\boldsymbol{w}}}\,\sum_{\boldsymbol{l}=\boldsymbol{\mu}_{\boldsymbol{w}}}^{\boldsymbol{\alpha}{-}\boldsymbol{j}{-}\boldsymbol{\mu}_{\boldsymbol{w}}}\,(\alpha_{1}{-}j_{1}{-}l_{1})\,j_{2}\,l_{3}\,w_{\boldsymbol{\alpha}{-}\boldsymbol{j}{-}\boldsymbol{l}}\,w_{\boldsymbol{j}}\,w_{\boldsymbol{l}}$
(110)

which reduces to eq. (40) in the case of
$\boldsymbol{\mu}_{\boldsymbol{w}}=\boldsymbol{0}$.

Going through the derivation for various values of $n$ allows us to realize
the pattern and find the general expression for $R$ which is given in the
following.

#### General expression for $R$

For $R$ in the $n$-variate case, we may write

$\displaystyle
R(\boldsymbol{\alpha},\boldsymbol{\beta})=\sum_{k=0}^{n}R_{n,k}$ (111)

where $R_{n,k}$ contains the terms with $k$ $w$-factors and correspondingly
$k{-}1$ sums. We find it convenient to use distinct expressions for $R_{n,k}$
for $k=0$, $k=1$, and999We could in principle get the $k=1$ expression as a
special case of eq. (115). $k\geq 2$. Defining

$\displaystyle\lambda_{i}:=\beta_{i}+v_{i}$ (112)

we have

$\displaystyle R_{n,0}$
$\displaystyle=\Big{(}\prod_{i=1}^{n}\lambda_{i}\Big{)}\delta_{\boldsymbol{\alpha},\boldsymbol{0}}$
(113) $\displaystyle R_{n,1}$
$\displaystyle=\bigg{(}\Big{(}\prod_{i=1}^{n}(\lambda_{i}{+}\alpha_{i})\Big{)}-\Big{(}\prod_{i=1}^{n}\lambda_{i}\Big{)}\bigg{)}w_{\boldsymbol{\alpha}}$
(114) $\displaystyle R_{n,k}$
$\displaystyle=\\!\\!\\!\sum_{\boldsymbol{i}_{\boldsymbol{2}}=\boldsymbol{\mu}_{\boldsymbol{w}}}^{\boldsymbol{\alpha}-(k-1)\boldsymbol{\mu}_{\boldsymbol{w}}}\cdots\\!\\!\\!\\!\\!\\!\sum_{\boldsymbol{i}_{\boldsymbol{j}}=\boldsymbol{\mu}_{\boldsymbol{w}}}^{\boldsymbol{\alpha}-\sum_{q=2}^{j-1}\boldsymbol{i}_{\boldsymbol{q}}-(k+1-j)\boldsymbol{\mu}_{\boldsymbol{w}}}\\!\\!\\!\\!\\!\\!\cdots\sum_{\boldsymbol{i}_{\boldsymbol{k}}=\boldsymbol{\mu}_{\boldsymbol{w}}}^{\boldsymbol{\alpha}{-}\sum_{q=2}^{k-1}\boldsymbol{i}_{\boldsymbol{q}}-\boldsymbol{\mu}_{\boldsymbol{w}}}$
$\displaystyle\quad
S_{n,k}\Big{(}\Big{(}\boldsymbol{\alpha}{-}\sum_{q=2}^{k}\boldsymbol{i}_{\boldsymbol{q}}\Big{)},\boldsymbol{i}_{\boldsymbol{2}},\ldots,\boldsymbol{i}_{\boldsymbol{k}},\boldsymbol{\lambda}\Big{)}\,w_{(\boldsymbol{\alpha}{-}\sum_{q=2}^{k}\boldsymbol{i}_{\boldsymbol{q}})}w_{\boldsymbol{i}_{\boldsymbol{2}}}\cdots
w_{\boldsymbol{i}_{\boldsymbol{k}}}$ (115)

where we have defined

$\displaystyle
S_{n,k}(\boldsymbol{i}_{\boldsymbol{1}},\boldsymbol{i}_{\boldsymbol{2}},\ldots,\boldsymbol{i}_{\boldsymbol{k}},\boldsymbol{\lambda})$
$\displaystyle:=\\!\\!\\!\\!\sum_{s\in\text{KSN}(n,k)}\\!\\!\bigg{(}\prod_{l\in\text{CS}(n,s)}\\!\\!\\!\\!\\!\\!\lambda_{l}\,\bigg{)}\prod_{y=1}^{k}\prod_{j\in
s_{y}}i_{y,j}$ (116)

Here we have $\text{KSN}(n,k)$ being defined as the set of all sets with size
$k$ of (non-empty and non-overlapping) subsets of the set of integers from 1
to $n$. An example being

$\displaystyle\text{KSN}(3,2)$
$\displaystyle=\Big{\\{}\big{\\{}\\{1\\},\\{2\\}\big{\\}},\big{\\{}\\{1\\},\\{3\\}\big{\\}},\big{\\{}\\{2\\},\\{3\\}\big{\\}},$
$\displaystyle\qquad\big{\\{}\\{1\\},\\{2,3\\}\big{\\}},\big{\\{}\\{2\\},\\{1,3\\}\big{\\}},\big{\\{}\\{3\\},\\{1,2\\}\big{\\}}\Big{\\}}$
(117)

where it should be possible to see the structure reflected in the single-sum
term of eq. (110). The other set-function $\text{CS}(n,s)$ gives the counter-
set to one of these sets, that is

$\displaystyle\text{CS}(n,s):=\\{1,\ldots,n\\}/\bigcup_{i}s_{i}$ (118)

with an example being

$\displaystyle\text{CS}(3,\big{\\{}\\{1\\},\\{3\\}\big{\\}})$
$\displaystyle=\\{2\\}$ (119)

We should point out that it is possible to write eq. (115) nicer and more
symmetrically at the cost of introducing an extra sum and a delta function,
that is101010 For practical implementations one can use finite upper limits
and replace
$\boldsymbol{\infty}\rightarrow\boldsymbol{\alpha}-(k{-}1)\boldsymbol{\mu}_{\boldsymbol{w}}$.

$\displaystyle R_{n,k}$
$\displaystyle=\\!\\!\\!\sum_{\boldsymbol{i}_{\boldsymbol{1}}=\boldsymbol{\mu}_{\boldsymbol{w}}}^{\boldsymbol{\infty}}\\!\\!\cdots\\!\\!\sum_{\boldsymbol{i}_{\boldsymbol{k}}=\boldsymbol{\mu}_{\boldsymbol{w}}}^{\boldsymbol{\infty}}\delta_{(\sum_{j}\\!\boldsymbol{i}_{\boldsymbol{j}}),\boldsymbol{\alpha}}\,S_{n,k}(\boldsymbol{i}_{\boldsymbol{1}},\ldots,\boldsymbol{i}_{\boldsymbol{k}},\boldsymbol{\lambda})\,w_{\boldsymbol{i}_{\boldsymbol{1}}}\cdots
w_{\boldsymbol{i}_{\boldsymbol{k}}}$ (120)

where evaluating the $\boldsymbol{i}_{\boldsymbol{1}}$-sum would bring back
eq. (115).

###  A.2 Algebraic expression in the rescaling frameworks

In Section 2.4 it was discussed how to simplify the computation of the
intersection number by rescaling the quantities on which it depends, locally
at each intersection point. Two different cases were presented, right
rescaling and left rescaling, which we will discuss below.

#### Right rescaling

Right rescaling is defined by the rescalings and renamings given in eq. (27)

$\displaystyle\hat{\varphi}_{L}\rightarrow\hat{\phi}=\hat{\varphi}_{L}\hat{\varphi}_{R}\,,\qquad\hat{\varphi}_{R}\rightarrow
1\,,\qquad u\rightarrow u_{R}=u/\hat{\varphi}_{R}\,,\qquad\psi\rightarrow
f\,.$ (121)

Eq. (104) may then be used for these rescaled variables, with
$\hat{\varphi}_{L,\boldsymbol{i}}$, $\hat{\varphi}_{R,\boldsymbol{i}}$,
$w_{\boldsymbol{i}}$, and $v_{j}$ interpreted as the coefficients in the
expansions of the rescaled variables $\hat{\phi}$, $1$, and $\log(u_{R})$, and
$\boldsymbol{\mu}_{\boldsymbol{L}}$, $\boldsymbol{\mu}_{\boldsymbol{R}}$,
$\boldsymbol{\mu}_{\boldsymbol{w}}$ as the corresponding powers of the leading
coefficients. But the expansion of $1$ is of course trivial, corresponding to
$\hat{\varphi}_{R,\boldsymbol{i}}=\delta_{\boldsymbol{i},\boldsymbol{0}}$ and
$\boldsymbol{\mu}_{\boldsymbol{R}}=0$. This allows us to do the $h$-sum of eq.
(104) giving

$\displaystyle\text{Res}(f)\;=\\!\\!\\!\\!\\!\sum_{\boldsymbol{\tau}=\boldsymbol{0}}^{n\boldsymbol{\mu}_{\boldsymbol{w}}-\boldsymbol{\mu}_{\boldsymbol{L}}-\boldsymbol{2}}\\!\\!\\!\\!\sum_{\sigma\in\text{VC}(\boldsymbol{\tau})}\\!\\!\\!\\!\\!Y(-\boldsymbol{\tau}{-}\boldsymbol{1},\boldsymbol{\tau},\sigma)\;\,\hat{\phi}_{n\boldsymbol{\mu}_{\boldsymbol{w}}-\boldsymbol{\tau}-\boldsymbol{2}}$
(122)

and inserting the expression for $Y$ the residue becomes

$\displaystyle\text{Res}(f)\;=\\!\\!\\!\\!\\!\sum_{\boldsymbol{\tau}=\boldsymbol{0}}^{n\boldsymbol{\mu}_{\boldsymbol{w}}-\boldsymbol{\mu}_{\boldsymbol{L}}-\boldsymbol{2}}\\!\\!\\!\\!\sum_{\sigma\in\text{VC}(\boldsymbol{\tau})}\frac{(-1)^{|\sigma|}}{U(\boldsymbol{0},\boldsymbol{0})}\prod_{i=1}^{|\sigma|}\frac{U(\boldsymbol{\tau}{-}\sum_{j<i}\boldsymbol{\sigma}_{\boldsymbol{j}},\,\boldsymbol{\sigma}_{\boldsymbol{i}})}{U(\boldsymbol{\tau}{-}\sum_{j<i}\boldsymbol{\sigma}_{\boldsymbol{j}},\,\boldsymbol{0})}\;\,\hat{\phi}_{n\boldsymbol{\mu}_{\boldsymbol{w}}{-}\boldsymbol{\tau}{-}\boldsymbol{2}}$
(123)

where

$\displaystyle U(\boldsymbol{\alpha},\boldsymbol{\beta})$
$\displaystyle=W(-\boldsymbol{\alpha}{-}\boldsymbol{1},\,\boldsymbol{\beta})$
$\displaystyle=R(\boldsymbol{\beta}{+}n\boldsymbol{\mu}_{\boldsymbol{w}},\,-\boldsymbol{\alpha}{-}\boldsymbol{1})$
(124)

Inserting $\boldsymbol{\mu}_{\boldsymbol{w}}=\boldsymbol{0}$, eq. (123)
reduces to the algebraic expression discussed in the main text in eq. (36).

#### Left rescaling

One may go through the same steps for the left rescaling framework, which is
defined in terms of the renamings and rescalings defined in eq. (29)

$\displaystyle\hat{\varphi}_{L}\rightarrow
1\,,\qquad\hat{\varphi}_{R}\rightarrow\hat{\phi}=\hat{\varphi}_{L}\hat{\varphi}_{R}\,,\qquad
u\rightarrow u_{L}=u\hat{\varphi}_{L}\,,\qquad\psi\rightarrow g\,.$ (125)

Inserting this in eq. (104), we get

$\displaystyle\text{Res}(g\hat{\varphi}_{L}\hat{\varphi}_{R})\;=\\!\\!\\!\sum_{\boldsymbol{\tau}=\boldsymbol{0}}^{n\boldsymbol{\mu}_{\boldsymbol{w}}-\boldsymbol{\mu}_{\boldsymbol{R}}-\boldsymbol{2}\;}\\!\\!\\!\\!\sum_{\sigma\in\text{VC}(\boldsymbol{\tau})}\frac{(-1)^{|\sigma|}}{\tilde{U}(\boldsymbol{\tau},\boldsymbol{0})}\prod_{i=1}^{|\sigma|}\frac{\tilde{U}(\sum_{j<i}\boldsymbol{\sigma}_{\boldsymbol{j}},\,\boldsymbol{\sigma}_{\boldsymbol{i}})}{\tilde{U}(\sum_{j<i}\boldsymbol{\sigma}_{\boldsymbol{j}},\,\boldsymbol{0})}\;\,\hat{\phi}_{n\boldsymbol{\mu}_{\boldsymbol{w}}{-}\boldsymbol{\tau}{-}\boldsymbol{2}}$
(126)

with

$\displaystyle\tilde{U}(\boldsymbol{\alpha},\boldsymbol{\beta})$
$\displaystyle=W(\boldsymbol{\alpha}{+}\boldsymbol{1}{-}n\boldsymbol{\mu}_{\boldsymbol{w}},\,\boldsymbol{\beta})$
$\displaystyle=R(\boldsymbol{\beta}{+}n\boldsymbol{\mu}_{\boldsymbol{w}},\,\boldsymbol{\alpha}{+}\boldsymbol{1}{-}n\boldsymbol{\mu}_{\boldsymbol{w}})$
(127)

Eq. (126) is the algebraic expression in the left rescaling framework.

###  A.3 Additional details

#### Expansion maxima

In eqs. (80, 81) we wrote the expansions of $\hat{\varphi}_{L}$,
$\hat{\varphi}_{R}$, and $w$ as formal expansions from the corresponding
$\boldsymbol{\mu}$ up to $\boldsymbol{\infty}$. Yet we do not need infinitely
many terms in the series, there is a maximal value above which the terms do
not contribute to the result of eq. (104). Considering eq. (104) as well as
the general $n$ expression for $R$ given in and below eq. (111) we may deduce
what these maximal values are. Using the notation $\boldsymbol{M}$ in parallel
with that used for the minimum values $\boldsymbol{\mu}$ we may write an
expansion only containing the terms that are actually needed:

$\displaystyle\hat{\varphi}_{L}=\sum_{\boldsymbol{i}=\boldsymbol{\mu}_{\boldsymbol{L}}}^{\boldsymbol{M}_{\boldsymbol{L}}}\\!\hat{\varphi}_{L,\boldsymbol{i}}z^{\boldsymbol{i}}\qquad\quad\hat{\varphi}_{R}=\sum_{\boldsymbol{i}=\boldsymbol{\mu}_{\boldsymbol{R}}}^{\boldsymbol{M}_{\boldsymbol{R}}}\\!\hat{\varphi}_{R,\boldsymbol{i}}z^{\boldsymbol{i}}\qquad\quad
w=\sum_{\boldsymbol{i}=\boldsymbol{\mu}_{\boldsymbol{w}}}^{\boldsymbol{M}_{\boldsymbol{w}}\\!}\\!w_{\boldsymbol{i}}z^{\boldsymbol{i}}\,+\,\sum_{j=1}^{n}v_{j}\log(z_{j})$
(128)

where we have

$\displaystyle\boldsymbol{M}_{\boldsymbol{L}}=n\boldsymbol{\mu}_{\boldsymbol{w}}{-}\boldsymbol{\mu}_{\boldsymbol{R}}{-}\boldsymbol{2}\,,\qquad\quad\boldsymbol{M}_{\boldsymbol{R}}=n\boldsymbol{\mu}_{\boldsymbol{w}}{-}\boldsymbol{\mu}_{\boldsymbol{L}}{-}\boldsymbol{2}\,,\qquad\quad\boldsymbol{M}_{\boldsymbol{w}}=(n{+}1)\boldsymbol{\mu}_{\boldsymbol{w}}{-}\boldsymbol{\mu}_{\boldsymbol{L}}{-}\boldsymbol{\mu}_{\boldsymbol{R}}{-}\boldsymbol{2}\,.$
(129)

This fact is useful for implementation purposes, since it tells which
coefficients to extract before applying the algebraic expression. Applying one
of the rescaled versions of the algebraic expression does not change these
limits if expressed in terms of the original $\varphi_{L}$ and $\varphi_{R}$.

#### The univariate case

In the univariate case there are two simplifications taking place which
deserve discussion. The first is that the vector compositions VC reduce to
standard compositions as discussed in Section 3.2. The second is that only one
$\omega$ is present which makes it desirable to express the result in terms of
expansions of that object, replacing the $w$-expansion of eq. (81) with its
separately treated log-terms. That is

$\displaystyle\omega=\sum_{i=\mu_{\omega}}^{\infty}\\!\omega_{i}\,z^{i}$ (130)

with the relation between the two expansions being
$\omega_{i}=(i{+}1)w_{i{+}1}$ except for $\omega_{-1}=v_{1}$. We also have
$\mu_{\omega}=\mu_{w}{-}1$. Inserting this in eq. (104) we get

$\displaystyle\text{Res}(\psi\varphi_{R})\;=\\!\\!\\!\\!\sum_{\tau=0}^{\mu_{\omega}-\mu_{\phi}-\mu_{\xi}-1\;}\\!\\!\\!\\!\sum_{\sigma\in\text{comp}(\tau)}\;\;\sum_{h=\mu_{\phi}-\mu_{\omega}}^{-\mu_{\xi}-\tau-1}\\!\\!\\!\\!\\!Y(h,\tau,\sigma)\;\hat{\varphi}_{L,h+\mu_{\omega}}\;\hat{\varphi}_{R,-h-\tau-1}$
(131)

where

$\displaystyle Y(h,\tau,\sigma)$
$\displaystyle=\frac{(-1)^{|\sigma|}}{W(h{+}\tau,0)}\prod_{i=1}^{|\sigma|}\frac{W(h{+}\sum_{j<i}\sigma_{j},\,\sigma_{i})}{W(h{+}\sum_{j<i}\sigma_{j},\,0)}$
(132)

as in eq. (105) but with

$\displaystyle
W(\eta,\lambda)=\eta\,\delta_{\lambda+\mu_{\omega},-1}+\omega_{\lambda+\mu_{\omega}}$
(133)

#### Further discussion

It is worthwhile to look at the special case in which everything (e.g.
$\varphi_{L}$, $\varphi_{R}$, and all $\omega_{i}$) has simple poles only.
This corresponds to
$\boldsymbol{\mu}_{\boldsymbol{L}}=\boldsymbol{\mu}_{\boldsymbol{R}}=\boldsymbol{-1}$
and $\boldsymbol{\mu}_{\boldsymbol{w}}=\boldsymbol{0}$. Inserting this in eq.
(104) we get that each of the sums contain one term only, leaving

$\displaystyle\text{Res}(\psi\varphi_{R})$
$\displaystyle=\frac{\hat{\varphi}_{L,\boldsymbol{-1}}\,\hat{\varphi}_{R,\boldsymbol{-1}}}{\prod_{i=1}^{n}v_{i}}$
(134)

Combining this with eq. (13) we get an expression for the intersection number
similar to an expression from refs. matsumoto1998 ; Mizera:2017rqa .

Additionally it is worth noticing that that the intersection number is linear
in both $\varphi_{L}$ and $\varphi_{R}$ which may be seen from the integral
definition in eq. (12). That property is easy to see in the algebraic
expression as it is given by eq. (104). But in the rescaling framework
linearity becomes hidden. For instance looking at the solution in the right-
rescaling framework as it is given by eq. (36), linearity in $\varphi_{L}$ is
obvious. Linearity in $\varphi_{R}$, however, becomes obscured as the
$\varphi_{R,\boldsymbol{i}}$-terms mix with the $w_{\boldsymbol{i}}$-factors,
which enter the expressions in a very non-linear fashion.

###  A.4 Comparing index vectors

(a) $\;\boldsymbol{a}\geq\boldsymbol{b}$

(b) $\;\boldsymbol{a}>\boldsymbol{b}$

(c) $\;\boldsymbol{a}\leq\boldsymbol{b}$

(d) $\;\boldsymbol{a}\nleq\boldsymbol{b}$

Figure 3:  Illustration of some of the binary operators defined in Table 5.
The shaded areas cover the integer values for which the illustrated relation
is true.

$\boldsymbol{a}=\boldsymbol{b}$ |  $\forall i$: $a_{i}=b_{i}$
---|---
$\boldsymbol{a}\leq\boldsymbol{b}$ |  $\forall i$: $a_{i}\leq b_{i}$
$\boldsymbol{a}\geq\boldsymbol{b}$ |  $\forall i$: $a_{i}\geq b_{i}$
$\boldsymbol{a}<\boldsymbol{b}$ |  $\boldsymbol{a}\leq\boldsymbol{b}$ and $\boldsymbol{a}\neq\boldsymbol{b}$

$\boldsymbol{a}\neq\boldsymbol{b}$ | $\text{not}(\boldsymbol{a}=\boldsymbol{b})$
---|---
$\boldsymbol{a}\nleq\boldsymbol{b}$ | $\text{not}(\boldsymbol{a}\leq\boldsymbol{b})$
$\boldsymbol{a}\ngeq\boldsymbol{b}$ | $\text{not}(\boldsymbol{a}\geq\boldsymbol{b})$
$\boldsymbol{a}>\boldsymbol{b}$ |  $\boldsymbol{a}\geq\boldsymbol{b}$ and $\boldsymbol{a}\neq\boldsymbol{b}$

Table 5: The definitions needed to compare index vectors.

In this appendix and in Section 3 we used the notation for the $n$-variate
formula eq. (36) where the sums and products are parameterized by index
vectors written in bold, of length $n$ and containing integers. Such vectors
can be compared using the so-called product order, let us briefly review how
it is done. If we let $\boldsymbol{a}$ and $\boldsymbol{b}$ be two such index
vectors of length $n$, the relations we use in this paper are defined as in
Table 5, and some are illustrated in Figure 3. We see that $\nleq$ and $>$ are
not equivalent, for instance it is true that $\binom{1}{2}\nleq\binom{2}{0}$
while it is false that $\binom{1}{2}>\binom{2}{0}$. We notice that the index
vectors form a partially ordered set under the $\leq$ operator.

##  Appendix B Algebraic solution via the integral formula

Here we derive the algebraic expression of a given residue appearing in the
main formula (13) that utilises the series expansion method presented in
Section 2.3. Throughout this section we assume that the pole of the residue is
located at the origin $z=0$.

Using the same notation as in Appendix A for various vector indices and
expansions, we write the $z$-expanded cocycles as:

$\displaystyle\hat{\varphi}_{L}=\sum_{{\boldsymbol{i}}\geq{\boldsymbol{\mu_{L}}}}\varphi_{L,{\boldsymbol{i}}}\>z^{\boldsymbol{i}}\
,\quad\hat{\varphi}_{R}=\sum_{{\boldsymbol{i}}\geq{\boldsymbol{\mu_{R}}}}\varphi_{R,{\boldsymbol{i}}}\>z^{\boldsymbol{i}}\
.$ (135)

Examples of the domains of summation appearing here are shown in Figure 4a. In
the following we will also refer to such domains of $z$-expansions as supports
111111 Recall that the support of a Laurent series
$f=\sum_{{\boldsymbol{i}}\in\mathbb{Z}^{n}}=f_{\boldsymbol{i}}\>z^{\boldsymbol{i}}$
is the set
$\mathrm{supp}(f):=\\{{\boldsymbol{i}}\in\mathbb{Z}^{n}\>|\>f_{\boldsymbol{i}}\neq
0\\}$ . . Expansion of the twist $u$ takes the following form:

$\displaystyle
u(z)=u_{0}\>z^{\boldsymbol{\gamma}}\>\bigl{(}1+\sum_{{\boldsymbol{i}}\cdot{\boldsymbol{1}}\geq
1}u_{\boldsymbol{i}}\>z^{\boldsymbol{i}}\bigr{)}\equiv
u_{0}\>z^{\boldsymbol{\gamma}}\>\bigl{(}1+\tilde{u}(z)\bigr{)}\
,\quad\tilde{u}(z):=\sum_{{\boldsymbol{i}}\cdot{\boldsymbol{1}}\geq
1}u_{\boldsymbol{i}}\>z^{\boldsymbol{i}}\ ,$ (136)

where $u_{0}$ is a $z$-independent constant, the vector
${\boldsymbol{\gamma}}$ is a linear combination of exponents appearing in (6),
$\tilde{u}$ collects terms of the expansion that have at least one power of
$z$ inside, and we denote:

$\displaystyle{\boldsymbol{i}}\cdot{\boldsymbol{1}}=i_{1}+\ldots+i_{n}\ .$
(137)

We show the supports of $u$ and $\tilde{u}$ in Figure 5a and Figure 5b
respectively. In the following we implicitely assume an additional condition
${\boldsymbol{i}}\geq{\boldsymbol{j}}$ whenever we write
$({\boldsymbol{i}}-{\boldsymbol{j}})\cdot{\boldsymbol{1}}\geq m$ for some
vector shift ${\boldsymbol{j}}$.

(a)
$\mathrm{supp}(\hat{\varphi}_{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{1.0}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{1.0}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{1.0}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{1.0}{$\scriptscriptstyle\bullet$}}}}}})$

(b) $\rho_{0}$

(c) $\rho_{1}$

(d) $\rho_{2}$

Figure 4:  Supports of the $\hat{\varphi}_{L}$ and $\hat{\varphi}_{R}$
expansions (135) in LABEL:sub@fig:cone-lr, and the corresponding summation
domains for eqs. (140, 143) in (LABEL:sub@fig:cone-rho0, LABEL:sub@fig:cone-
rho1, LABEL:sub@fig:cone-rho2).

(a) $\mathrm{supp}(u)$

(b) $\mathrm{supp}(\tilde{u})$

(c) $\mathrm{supp}(\tilde{u}^{2})$

(d) for eq. (143)

Figure 5:  Supports of the expansions (136) in (LABEL:sub@fig:cone-u0,
LABEL:sub@fig:cone-u1, LABEL:sub@fig:cone-u2), where we implicitely assume an
extra ${\boldsymbol{i}}\geq{\boldsymbol{0}}$ condition. In
(LABEL:sub@fig:cone-u-1) we illustrate the notation of eq. (143) (an
additional ${\boldsymbol{i}}\leq{\boldsymbol{0}}$ here is implied as well).

According to the integral formula (22), our goal is to compute the following
residue:

$\displaystyle\rho:={\rm
Res}_{z=0}\Bigl{[}\varphi_{R}u^{-1}\int^{z}\varphi_{L}u\Bigr{]}$ (138)

The key idea is to expand the inverse of the twist $u(z)^{-1}$ in terms of
$\tilde{u}(z)$ using the geometric series 121212 For brevity we omit the
$z^{-\gamma}$ and $z^{\gamma}$ factors around the integral sign that come from
the twist (136), but they can easily be restored via: $\int\varphi\mapsto
z^{-\gamma}\int^{z}z^{\gamma}\varphi$ (with some abuse of notation). :

$\displaystyle\rho$ $\displaystyle=\underbrace{{\rm
Res}_{z=0}\Bigl{[}\varphi_{R}\int\varphi_{L}\Bigr{]}}_{\rho_{0}}+\sum_{m=1}^{(-{\boldsymbol{\mu_{L}}}-{\boldsymbol{\mu_{R}}}-{\boldsymbol{2}})\cdot{\boldsymbol{1}}}(-1)^{m}\>\underbrace{{\rm
Res}_{z=0}\Bigl{[}\varphi_{R}\cdot\tilde{u}^{m-1}\cdot\Bigl{(}\tilde{u}\int\varphi_{L}-\int\tilde{u}\varphi_{L}\Bigr{)}\Bigr{]}}_{\rho_{m}}\
,$ (139)

where in $\rho_{m}$ we gather the residues with $m$ insertions of
$\tilde{u}(z)$. The summation bounds in the second term follows from the fact
that the $z$-expansion of $\tilde{u}(z)^{m}$ has terms with at least $m$
powers of $z$ inside, as reflected in Figure 5b and Figure 5c for $m=1$ and
$m=2$ respectively.

The residue $\rho_{0}$ in eq. (139) is straightforward to evaluate:

$\displaystyle\rho_{0}=\sum_{{\boldsymbol{i}}={\boldsymbol{\mu_{L}}}+{\boldsymbol{1}}}^{-{\boldsymbol{\mu_{R}}}-{\boldsymbol{1}}}\hat{\varphi}_{R,-{\boldsymbol{i}}-{\boldsymbol{1}}}\>\hat{\varphi}_{L,{\boldsymbol{i}}-{\boldsymbol{1}}}\>\frac{1}{{\boldsymbol{i}}+{\boldsymbol{\gamma}}}\
,$ (140)

where the fraction stands for
$1/({\boldsymbol{i}}+{\boldsymbol{\gamma}}):=1/\prod_{j}(i_{j}+\gamma_{j})$ ,
and the $n$-dimensional summation domain is shown in Figure 4b.

To compute $\rho_{m}$ in eq. (139) we separately expand its three factors,
multiply them, and take the residue. The first factor is given in eq. (135).
The second factor is a power of a series, so we may write its
${\boldsymbol{i}}$th non-zero coefficient
$\tilde{u}(z)^{m}=\sum_{\boldsymbol{i}}(\tilde{u}^{m})_{\boldsymbol{i}}\>z^{\boldsymbol{i}}$
using the Kronecker symbol $\delta_{{\boldsymbol{i}},{\boldsymbol{j}}}$ as:

$\displaystyle\bigl{(}\tilde{u}^{m}\bigr{)}_{\boldsymbol{i}}=\sum_{{{\boldsymbol{j_{1}}}\cdot{\boldsymbol{1}}}\geq
1}\ldots\sum_{{{\boldsymbol{j_{m}}}}\cdot{\boldsymbol{1}}\geq
1}\delta_{(\sum_{k}{\boldsymbol{j_{k}}}),{\boldsymbol{i}}}\>u_{\boldsymbol{j_{1}}}\ldots
u_{\boldsymbol{j_{m}}}\ ,\quad\text{for
${{{\boldsymbol{i}}\cdot{\boldsymbol{1}}}\geq m}$ and
${\boldsymbol{i}}\geq{\boldsymbol{0}}$}\ .$ (141)

One can come up with various evaluation strategies for this multiple sum
using, for example, the nested parametrization as in eq. (115), or the
(precomputed) multivariate Bell polynomials Withers:2008 . For the
${\boldsymbol{i}}$th non-zero coefficient of the last factor of $\rho_{m}$ we
get

$\displaystyle\Bigl{(}\tilde{u}\int\varphi_{L}-\int\tilde{u}\varphi_{L}\Bigr{)}_{\boldsymbol{i}}=\sum_{\begin{subarray}{c}{{\boldsymbol{j}}\cdot{\boldsymbol{1}}}\geq
1\\\
{\boldsymbol{j}}\leq{\boldsymbol{i}}-{\boldsymbol{\mu_{L}}}-{\boldsymbol{1}}\end{subarray}}u_{\boldsymbol{j}}\>\hat{\varphi}_{L,{\boldsymbol{i}}-{\boldsymbol{j}}-{\boldsymbol{1}}}\>\Bigl{(}\tfrac{1}{{\boldsymbol{i}}-{\boldsymbol{j}}+{\boldsymbol{\gamma}}}-\tfrac{1}{{\boldsymbol{i}}+{\boldsymbol{\gamma}}}\Bigr{)}\
,\quad\text{for
${({\boldsymbol{i}}-{\boldsymbol{\mu_{L}}}-{\boldsymbol{1}})\cdot{\boldsymbol{1}}\geq
1}$}$ (142)

and ${\boldsymbol{i}}\geq{\boldsymbol{\mu_{L}}}+{\boldsymbol{1}}$. Integrals
here are done via the same technique as discussed in Section 2.3.

Finally the sum representation of the $\rho_{m}$ residue reads:

$\displaystyle\rho_{m}=\sum_{\begin{subarray}{c}({\boldsymbol{i}}-{\boldsymbol{\mu_{L}}}-{\boldsymbol{1}})\cdot{\boldsymbol{1}}\geq
m\\\
{\boldsymbol{i}}\leq-{\boldsymbol{\mu_{R}}}-{\boldsymbol{1}}\end{subarray}}\hat{\varphi}_{R,-{\boldsymbol{i}}-{\boldsymbol{1}}}\sum_{\begin{subarray}{c}{{\boldsymbol{j}}}\cdot{\boldsymbol{1}}\geq
m-1\\\
({\boldsymbol{j}}-{\boldsymbol{i}}+{\boldsymbol{\mu_{L}}}+{\boldsymbol{1}})\cdot{\boldsymbol{1}}\leq-1\end{subarray}}(\tilde{u}^{m-1})_{\boldsymbol{j}}\Bigl{(}\tilde{u}\int\varphi_{L}-\int\tilde{u}\varphi_{L}\Bigr{)}_{{\boldsymbol{i}}-{\boldsymbol{j}}}\
,$ (143)

where examples of the outer sum’s domain are shown in figures 4c and 4d (see
also Figure 5d for an illustration of the $\leq$ notation in the inner sum).
Alongside the non-vanishing condition:

$\displaystyle-{\boldsymbol{\mu_{L}}}-{\boldsymbol{\mu_{R}}}-{\boldsymbol{2}}\geq{\boldsymbol{0}}\
,$ (144)

we obtain the algebraic expression for a given residue:

$\displaystyle\boxed{{\rm
Res}(\psi\varphi_{R})=\sum_{m=0}^{(-{\boldsymbol{\mu_{L}}}-{\boldsymbol{\mu_{R}}}-{\boldsymbol{2}})\cdot{\boldsymbol{1}}}(-1)^{m}\rho_{m}}$
(145)

with $\rho_{m}$ defined by eqs. (140) and (143, 141, 142).

##  Appendix C Additional details for the examples

In this appendix we provide further details for the examples presented in
Section 5.

###  C.1 Two-loop massless sunrise diagram

Here we elaborate on the material of Section 5.2, namely we discuss the
coordinate transformations presented in Table 2, focusing on the intersection
point $z=(s,0)$ as an illustrative example.

The main purpose of our coordinate transformations is to compute the
multivariate residue in the main formula (13) at the selected intersection
point. So we start with a shift $(z_{1},z_{2})\mapsto(s+z_{1},z_{2})$, after
which the twist (50) turns into:

$\displaystyle
u=(s+z_{1})^{\gamma_{1}}z_{2}^{\gamma_{2}}\>(z_{1}+z_{2})^{\gamma_{3}}\ .$
(146)

The last factor here is problematic: its series expansion around the new
origin $z_{1}=z_{2}=0$ is ill-defined. For example, expanding
$1/(z_{1}+z_{2})$ in the following two ways we get different results:

$\displaystyle
1/(z_{1}+z_{2})=\begin{cases}z_{2}^{-1}-z_{1}z_{2}^{-2}+\ldots&\text{first in
$z_{1}$, then in $z_{2}$\ ,}\\\ z_{1}^{-1}-z_{1}^{-2}z_{2}+\ldots&\text{first
in $z_{2}$, then in $z_{1}$\ .}\end{cases}$ (147)

To fix this we further change $(z_{1},z_{2})\mapsto(z_{1},z_{1}z_{2})$, so
that the twist (146) becomes regular:

$\displaystyle
u=z_{1}^{\gamma_{2}+\gamma_{3}}z_{2}^{\gamma_{2}}(s+z_{1})^{\gamma_{1}}\>(1+z_{2})^{\gamma_{3}}\
,$ (148)

meaning that all the polynomial factors now have constant terms. Series
expansion of eq. (148) no longer depends on the order of operations and we may
proceed with evaluation of the multivariate residue in eq. (13) as discussed
in Section 2 and Section 5.2. Putting together the transformations that led us
to the twist (148) gives:

$\displaystyle(z_{1},z_{2})\mapsto(s+z_{1},z_{1}z_{2})\ ,$ (149)

where we recognize one of the elements listed on the second row of Table 2.

In general, coordinate transformations such as (149) are also known as
resolutions of singularities of algebraic varieties. They can be computed
algorithmically with the help of, for example, pySecDec Borowka:2017idc ;
Heinrich:2021dbf .

###  C.2 Two-loop planar box diagram

Chart | Point | Coordinate transformation(s)
---|---|---
$U_{z}$ | $(0,0)$ | $(z_{1},\>z_{2})$
$(-t,0)$ | $(-t+(s-t)\sum_{i=1}^{N}(-z_{1}z_{2}/s)^{i}+z_{1}(z_{1}z_{2})^{N}\>,\>z_{1}z_{2})$
$(0,-t)$ | $(z_{1}z_{2}\>,\>-t+(s-t)\sum_{i=1}^{N}(-z_{1}z_{2}/s)^{i}+(z_{1}z_{2})^{N}z_{2})$
$U_{x}$ | $(0,0)$ | $\bigl{\\{}(z_{1},\tfrac{1}{z_{2}}),\>(\tfrac{1}{z_{2}},\tfrac{1}{z_{1}z_{2}}),\>(-s-(s-t)\sum_{i=1}^{N}(-s\>z_{1}z_{2})^{i}+z_{1}(z_{1}z_{2})^{N}\>,\>\tfrac{1}{z_{1}z_{2}})\bigr{\\}}$
$U_{y}$ | $(0,0)$ | $\bigl{\\{}(\tfrac{1}{z_{1}z_{2}},\tfrac{1}{z_{1}}),\>(\tfrac{1}{z_{1}},z_{2}),\>(\tfrac{1}{z_{1}z_{2}}\>,\>-s-(s-t)\sum_{i=1}^{N}(-s\>z_{1}z_{2})^{i}+(z_{1}z_{2})^{N}z_{2})\bigr{\\}}$
Table 6:  Intersection points (middle) from the three coordinate charts of
$\mathbb{C}\mathrm{P}^{2}$ (left) that contribute to the intersection numbers
between forms (52) with the massless 2-loop planar box twist (64). On the
right we show the coordinate transformations used to compute the residues
(13). Some transformations are parametrized by $N\in\mathbb{Z}_{\geq 0}$ , see
Appendix C.2. We sum over all the contributions of each displayed
transformation.

Here we provide further details for Section 5.3, in particular in Table 6 we
collect the coordinate transformations needed for evaluation of intersection
numbers between forms (67) with the twist (64).

Consider the point $z=(-t,0)$ appearing on the second row of Table 6. This
point is located at the intersection of the two singular hypersurfaces
$\mathcal{S}_{2}\cap\mathcal{S}_{3}$, i.e. satisfies
$\mathcal{B}_{2}=\mathcal{B}_{3}=0$ (see eq. (65)). We look for a pair of
local coordinates that roughly looks like
$(z_{1},z_{2})\mapsto(\mathcal{B}_{3}(z_{1},z_{2}),\mathcal{B}_{2}(z_{1},z_{2}))$
. Since $\mathcal{B}_{2}(z_{1},z_{2}):=z_{2}$ we may keep $z_{2}$ as one of
our new coordinates. To derive the transformation rule for $z_{1}$ solve:

$\displaystyle\mathcal{B}_{3}(z_{1},z_{2}):=st+s(z_{1}+z_{2})+z_{1}z_{2}=0\>\Longrightarrow\>z_{1}$
$\displaystyle=-t+(s-t)\>\frac{-z_{2}/s}{1+z_{2}/s}\ .$ (150)

and expand this solution near $z_{2}=0$ producing:

$\displaystyle
z_{1}=-t+(s-t)\bigl{(}-z_{2}/s+(-z_{2}/s)^{2}+\ldots+(-z_{2}/s)^{N}+\ldots\bigr{)}\
.$ (151)

Truncating this expansion up to the $N\textsuperscript{th}$ order and an
additional change $(z_{1},z_{2})\mapsto(z_{1},z_{1}z_{2})$ reproduces (up to
the $z_{1}(z_{1}z_{2})^{N}$ term) the coordinate transformation shown on the
second row of Table 6.

In these new coordinates, the twist becomes:

$\displaystyle
u=z_{1}^{\gamma_{2}+(N+1)\gamma_{3}}z_{2}^{\gamma_{2}+N\gamma_{3}}(-t+\ldots+z_{1}(z_{1}z_{2})^{N})^{\gamma_{1}}\bigl{(}s-(s-t)s^{-N}\>z_{2}+z_{1}z_{2}\bigr{)}^{\gamma_{3}}\
,$ (152)

where each polynomial factor now has a constant term, so that the series
expansion around $z_{1}=z_{2}=0$ is well-defined. A generic monomial 2-form
(67) transforms as follows:

$\displaystyle\varphi^{(\mathbf{2})}_{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{1.0}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{1.0}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{1.0}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{1.0}{$\scriptscriptstyle\bullet$}}}}}}:=z_{1}^{n_{1}}z_{2}^{n_{2}}\>\mathrm{d}z_{1}\wedge\mathrm{d}z_{2}\mapsto
z_{1}^{N+1+n_{2}}z_{2}^{N+n_{2}}\>(-t+\ldots+z_{1}(z_{1}z_{2})^{N})^{n_{1}}\>\mathrm{d}z_{1}\wedge\mathrm{d}z_{2}\
.$ (153)

The pole order of the form shifts by $N$ in both variables, which implies that
for large enough $N$ the non-vanishing condition (144) will no longer be
satisfied and the corresponding transformation rule will not contribute to the
intersection number. Hence we conclude, that even though the Table 6 collects
an infinite number of coordinate transformations, only a finite number of them
contributes to a given intersection number.

##  Appendix D Relation to Matsumoto’s algorithm

In this appendix we show how the partial differential equation in (21) for the
case of $n=2$ is indeed compatible with the results of matsumoto1998 .

The derivation of the intersection number for 2-forms presented in
matsumoto1998  involves the 2-form $\varphi_{I}$, the 1-forms $\psi_{P}^{p}$
and $\psi_{P}^{q}$, and the 0-form $\psi_{P}^{P}$, which are shown to satisfy
the following equations:

$\displaystyle\nabla_{\omega}\psi_{P}^{p}=\nabla_{\omega}\psi_{P}^{q}=\varphi_{I}\
,$ (154) $\displaystyle\nabla_{\omega}\psi_{P}^{P}=\psi_{P}^{p}-\psi_{P}^{q}\
,$ (155)

where $\nabla_{\omega}=\nabla_{\omega_{1}}+\nabla_{\omega_{2}}$ . We stress
that owing to this system of equations, the scalar function $\psi_{P}^{P}$ is
determined from the last equation that requires the knowledge of two 1-forms,
appearing in the non-homogenous term, that, in turn, are determined by solving
the former two equations. Along the derivation of the last relation, the
following auxiliary equations are employed,

$\displaystyle\nabla_{\omega_{2}}\psi_{P}^{P}=\psi_{P}^{p}\ ,$ (156)

and

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\nabla_{\omega}\Bigl{(}\nabla_{\omega}\psi_{P}^{P}-(\psi_{P}^{p}-\psi_{P}^{q})\Bigr{)}=\nabla_{\omega}\Bigl{(}\nabla_{\omega_{1}}\psi_{P}^{P}+\psi_{P}^{q}\Bigr{)}$
(157)

to deduce:

$\displaystyle\nabla_{\omega_{1}}\psi_{P}^{P}=-\psi_{P}^{q}\ .$ (158)

In order to recognize that the above set of equations imply eq. (21), let us
rename: the 2-form $\varphi_{I}\equiv\varphi_{L}^{({\bf 2})}$; the 1-forms
$\psi_{P}^{p}\equiv\psi^{(1)}$ and $\psi_{P}^{q}\equiv\psi^{(2)}$; and the
scalar 0-form $\psi_{P}^{P}\equiv-\psi$. Therefore, the five equations above
read, respectively:

$\displaystyle\nabla_{\omega}\psi^{(1)}$ $\displaystyle=$
$\displaystyle\nabla_{\omega}\psi^{(2)}=\varphi_{L}^{({\bf 2})}$ (159)
$\displaystyle\nabla_{\omega}\psi$ $\displaystyle=$
$\displaystyle\psi^{(2)}-\psi^{(1)}$ (160)
$\displaystyle\nabla_{\omega_{2}}\psi$ $\displaystyle=$
$\displaystyle-\psi^{(1)}$ (161) $\displaystyle 0$ $\displaystyle=$
$\displaystyle\nabla_{\omega}\Bigl{(}\nabla_{\omega_{1}}(-\psi)+\psi^{(2)}\Bigr{)}$
(162) $\displaystyle\nabla_{\omega_{1}}\psi$ $\displaystyle=$
$\displaystyle\psi^{(2)}$ (163)

The penultimate equation can be made explicit, to recognise finally a (hidden)
equation:

$\displaystyle 0$ $\displaystyle=$
$\displaystyle-\nabla_{\omega}\nabla_{\omega_{1}}\psi+\nabla_{\omega}\psi^{(2)}=-\nabla_{\omega_{2}}\nabla_{\omega_{1}}\psi+\varphi_{L}^{({\bf
2})}\ .$ (164)

The latter amounts to eq. (21) for 2-forms, yielding a straightforward
determination of the scalar function $\psi$ just from one equation. $\square$

## References

  * (1) K. Matsumoto, Quadratic Identities for Hypergeometric Series of Type $(k,l)$, Kyushu Journal of Mathematics 48 (1994), no. 2 335–345.
  * (2) K. Matsumoto, Intersection numbers for logarithmic $k$-forms, Osaka J. Math. 35 (1998), no. 4 873–893.
  * (3) K. Ohara, Y. Sugiki, and N. Takayama, Quadratic Relations for Generalized Hypergeometric Functions ${}_{p}F_{p-1}$, Funkcialaj Ekvacioj 46 (2003), no. 2 213–251.
  * (4) Y. Goto, Twisted Cycles and Twisted Period Relations for Lauricella’s Hypergeometric Function $F_{C}$, International Journal of Mathematics 24 (2013), no. 12 1350094, [arXiv:1308.5535].
  * (5) Y. Goto and K. Matsumoto, The monodromy representation and twisted period relations for Appell’s hypergeometric function $F_{4}$, Nagoya Math. J. 217 (03, 2015) 61–94.
  * (6) Y. Goto, Twisted period relations for Lauricella’s hypergeometric functions $F_{A}$, Osaka J. Math. 52 (07, 2015) 861–879.
  * (7) Y. Goto, Intersection Numbers and Twisted Period Relations for the Generalized Hypergeometric Function ${}_{m+1}F_{m}$, Kyushu Journal of Mathematics 69 (2015), no. 1 203–217.
  * (8) S. Mizera, Scattering Amplitudes from Intersection Theory, Phys. Rev. Lett. 120 (2018), no. 14 141602, [arXiv:1711.00469].
  * (9) S.-J. Matsubara-Heo and N. Takayama, An algorithm of computing cohomology intersection number of hypergeometric integrals, Nagoya Mathematical Journal (2019) 1–17, [arXiv:1904.01253].
  * (10) K. Ohara, “Intersection numbers of twisted cohomology groups associated with Selberg-type integrals.” http://www.math.kobe-u.ac.jp/HOME/ohara/Math/980523.ps, 1998.
  * (11) Y. Goto and S.-J. Matsubara-Heo, Homology and cohomology intersection numbers of gkz systems, 2020.
  * (12) S.-J. Matsubara-Heo, Computing cohomology intersection numbers of gkz hypergeometric systems, 2020.
  * (13) S.-J. Matsubara-Heo, Localization formulas of cohomology intersection numbers, 2021.
  * (14) P. Mastrolia and S. Mizera, Feynman Integrals and Intersection Theory, JHEP 02 (2019) 139, [arXiv:1810.03818].
  * (15) H. Frellesvig, F. Gasparotto, S. Laporta, M. K. Mandal, P. Mastrolia, L. Mattiazzi, and S. Mizera, Decomposition of Feynman Integrals on the Maximal Cut by Intersection Numbers, JHEP 05 (2019) 153, [arXiv:1901.11510].
  * (16) H. Frellesvig, F. Gasparotto, M. K. Mandal, P. Mastrolia, L. Mattiazzi, and S. Mizera, Vector Space of Feynman Integrals and Multivariate Intersection Numbers, Phys. Rev. Lett. 123 (2019), no. 20 201602, [arXiv:1907.02000].
  * (17) H. Frellesvig, F. Gasparotto, S. Laporta, M. K. Mandal, P. Mastrolia, L. Mattiazzi, and S. Mizera, Decomposition of Feynman Integrals by Multivariate Intersection Numbers, JHEP 03 (2021) 027, [arXiv:2008.04823].
  * (18) K. Aomoto and M. Kita, Theory of Hypergeometric Functions.  Springer Monographs in Mathematics. Springer Japan, 2011.
  * (19) S. Mizera and A. Pokraka, From Infinity to Four Dimensions: Higher Residue Pairings and Feynman Integrals, JHEP 02 (2020) 159, [arXiv:1910.11852].
  * (20) J. Chen, X. Jiang, X. Xu, and L. L. Yang, Constructing canonical Feynman integrals with intersection theory, Phys. Lett. B 814 (2021) 136085, [arXiv:2008.03045].
  * (21) S. Weinzierl, Applications of intersection numbers in physics, in MathemAmplitudes 2019: Intersection Theory and Feynman Integrals, 11, 2020\.  arXiv:2011.02865.
  * (22) S. Caron-Huot and A. Pokraka, Duals of Feynman integrals. Part I. Differential equations, JHEP 12 (2021) 045, [arXiv:2104.06898].
  * (23) S. Caron-Huot and A. Pokraka, Duals of Feynman Integrals. Part II. Generalized unitarity, JHEP 04 (2022) 078, [arXiv:2112.00055].
  * (24) J. Chen, X. Jiang, C. Ma, X. Xu, and L. L. Yang, Baikov representations, intersection theory, and canonical Feynman integrals, JHEP 07 (2022) 066, [arXiv:2202.08127].
  * (25) S. Mizera, Status of Intersection Theory and Feynman Integrals, PoS MA2019 (2019) 016, [arXiv:2002.10476].
  * (26) S. L. Cacciatori, M. Conti, and S. Trevisan, Co-Homology of Differential Forms and Feynman Diagrams, Universe 7 (2021), no. 9 328, [arXiv:2107.14721].
  * (27) S. Abreu, R. Britto, and C. Duhr, The SAGEX Review on Scattering Amplitudes, Chapter 3: Mathematical structures in Feynman integrals, arXiv:2203.13014.
  * (28) S. Weinzierl, Feynman Integrals, arXiv:2201.03593.
  * (29) P. Mastrolia, From Diagrammar to Diagrammalgebra, PoS MA2019 (2022) 015.
  * (30) H. A. Frellesvig and L. Mattiazzi, On the Application of Intersection Theory to Feynman Integrals: the univariate case, PoS MA2019 (2022) 017, [arXiv:2102.01576].
  * (31) M. K. Mandal and F. Gasparotto, On the Application of Intersection Theory to Feynman Integrals: the multivariate case, PoS MA2019 (2022) 019\.
  * (32) C. Ma, Y. Wang, X. Xu, L. L. Yang, and B. Zhou, Mixed QCD-EW corrections for Higgs leptonic decay via $HW^{+}W^{-}$ vertex, JHEP 09 (2021) 114, [arXiv:2105.06316].
  * (33) S. Abreu, R. Britto, C. Duhr, E. Gardi, and J. Matthew, From positive geometries to a coaction on hypergeometric functions, JHEP 02 (2020) 122, [arXiv:1910.08358].
  * (34) S. Abreu, R. Britto, C. Duhr, E. Gardi, and J. Matthew, Generalized hypergeometric functions and intersection theory for Feynman integrals, PoS (2019), no. RACOR2019 067, [arXiv:1912.03205].
  * (35) S. Abreu, R. Britto, C. Duhr, E. Gardi, and J. Matthew, The diagrammatic coaction beyond one loop, JHEP 10 (2021) 131, [arXiv:2106.01280].
  * (36) S. Abreu, R. Britto, C. Duhr, E. Gardi, and J. Matthew, The Diagrammatic Coaction, in 16th DESY Workshop on Elementary Particle Physics: Loops and Legs in Quantum Field Theory 2022, 7, 2022.  arXiv:2207.07843.
  * (37) S. Mizera, Combinatorics and Topology of Kawai-Lewellen-Tye Relations, JHEP 08 (2017) 097, [arXiv:1706.08527].
  * (38) A. Kaderli, A note on the Drinfeld associator for genus-zero superstring amplitudes in twisted de Rham theory, J. Phys. A 53 (2020), no. 41 415401, [arXiv:1912.09406].
  * (39) N. Kalyanapuram and R. G. Jha, Positive Geometries for all Scalar Theories from Twisted Intersection Theory, Phys. Rev. Res. 2 (2020), no. 3 033119, [arXiv:2006.15359].
  * (40) S. Weinzierl, Correlation functions on the lattice and twisted cocycles, Phys. Lett. B 805 (2020) 135449, [arXiv:2003.05839].
  * (41) K. Cho and K. Matsumoto, Intersection theory for twisted cohomologies and twisted Riemann’s period relations I, Nagoya Math. J. 139 (1995) 67–86.
  * (42) S. Mizera, Aspects of Scattering Amplitudes and Moduli Space Localization, arXiv:1906.02099.
  * (43) S. Weinzierl, On the computation of intersection numbers for twisted cocycles, J. Math. Phys. 62 (2021), no. 7 072301, [arXiv:2002.01930].
  * (44) K. Matsumoto, Relative twisted homology and cohomology groups associated with Lauricella’s $F_{D}$, arXiv:1804.00366.
  * (45) S.-J. Matsubara-Heo and N. Takayama, Algorithms for pfaffian systems and cohomology intersection numbers of hypergeometric integrals, in Lecture Notes in Computer Science, Lecture notes in computer science, pp. 73–84.  Springer International Publishing, 2020.  Errata in http://www.math.kobe-u.ac.jp/OpenXM/Math/intersection2/.
  * (46) V. Chestnov, F. Gasparotto, M. K. Mandal, P. Mastrolia, S. J. Matsubara-Heo, H. J. Munch, and N. Takayama, Macaulay Matrix for Feynman Integrals: Linear Relations and Intersection Numbers, arXiv:2204.12983.
  * (47) P. A. Baikov, Explicit solutions of the multiloop integral recurrence relations and its application, Nucl. Instrum. Meth. A389 (1997) 347–349, [hep-ph/9611449].
  * (48) E. Witten, Analytic Continuation Of Chern-Simons Theory, AMS/IP Stud. Adv. Math. 50 (2011) 347–446, [arXiv:1001.2933].
  * (49) R. N. Lee and A. A. Pomeransky, Critical points and number of master integrals, JHEP 11 (2013) 165, [arXiv:1308.6676].
  * (50) T. Bitoun, C. Bogner, R. P. Klausen, and E. Panzer, Feynman integral relations from parametric annihilators, Lett. Math. Phys. 109 (2019), no. 3 497–564, [arXiv:1712.09215].
  * (51) D. Agostini, C. Fevola, A.-L. Sattelberger, and S. Telen, Vector Spaces of Generalized Euler Integrals, arXiv:2208.08967.
  * (52) M. Søgaard and Y. Zhang, Multivariate Residues and Maximal Unitarity, JHEP 12 (2013) 008, [arXiv:1310.6006].
  * (53) G. E. Andrews, The Theory of Partitions.  Encyclopedia of Mathematics and its Applications. Cambridge University Press, 1984.
  * (54) F. V. Tkachov, A Theorem on Analytical Calculability of Four Loop Renormalization Group Functions, Phys. Lett. B 100 (1981) 65–68.
  * (55) K. G. Chetyrkin and F. V. Tkachov, Integration by Parts: The Algorithm to Calculate beta Functions in 4 Loops, Nucl. Phys. B192 (1981) 159–204.
  * (56) S. Laporta, High precision calculation of multiloop Feynman integrals by difference equations, Int. J. Mod. Phys. A15 (2000) 5087–5159, [hep-ph/0102033].
  * (57) S. Borowka, G. Heinrich, S. Jahn, S. P. Jones, M. Kerner, J. Schlenk, and T. Zirke, pySecDec: a toolbox for the numerical evaluation of multi-scale integrals, Comput. Phys. Commun. 222 (2018) 313–326, [arXiv:1703.09692].
  * (58) G. Heinrich, S. Jahn, S. P. Jones, M. Kerner, F. Langer, V. Magerya, A. Pöldaru, J. Schlenk, and E. Villa, Expansion by regions with pySecDec, Comput. Phys. Commun. 273 (2022) 108267, [arXiv:2108.10807].
  * (59) C. Bogner and S. Weinzierl, Resolution of singularities for multi-loop integrals, Comput. Phys. Commun. 178 (2008) 596–610, [arXiv:0709.4092].
  * (60) C. S. Withers and S. Nadarajah, Multivariate Bell polynomials and their applications to powers and fractionary iterates of vector power series and to partial derivatives of composite vector functions, Applied Mathematics and Computation 206 (2008), no. 2 997–1004.