# Locality Galois groups
of meromorphic germs in several variables

Li Guo  Department of Mathematics and Computer Science, Rutgers University,
Newark, NJ 07102, USA<EMAIL_ADDRESS>,  Sylvie Paycha  Institute of
Mathematics, University of Potsdam, D-14469 Potsdam, Germany<EMAIL_ADDRESS>potsdam.de and  Bin Zhang  School of Mathematics, Sichuan University, Chengdu,
610064, China<EMAIL_ADDRESS>
###### Abstract.

Meromorphic germs in several variables with linear poles naturally arise in
mathematics in various disguises. We investigate their rich structures under
the prism of locality, including locality subalgebras, locality transformation
groups and locality characters. The key technical tool is the dependence
subspace for a meromorphic germ with which we define a locality orthogonal
relation between two meromorphic germs. We describe the structure of locality
subalgebras generated by classes of meromorphic germs with certain types of
poles. We also define and determine their group of locality transformations
which fix the holomorphic germs and preserve multivariable residues, a group
we call the locality Galois group.

We then specialise to two classes of meromorphic germs with prescribed types
of nested poles, arising from multiple zeta functions in number theory and
Feynman integrals in perturbative quantum field theory respectively. We show
that they are locality polynomial subalgebras with locality polynomial bases
given by the locality counterpart of Lyndon words. This enables us to
explicitly describe their locality Galois group. As an application, we propose
a mathematical interpretation of Speer’s analytic renormalisation for Feynman
amplitudes. We study a class of locality characters, called generalised
evaluators after Speer. We show that the locality Galois group acts
transitively on generalised evaluators by composition, thus providing a
candidate for a renormalisation group in this multivariable approach.

###### Key words and phrases:

meromorphic germ, locality, Galois group, Lyndon word, renormalisation,
evaluator, convex cone, renormalisation group

###### 2010 Mathematics Subject Classification:

32A20, 13B05, 08A55, 32A27, 81T15, 15A63, 52C07, 81T17, 11M32, 05A05

###### Contents

  1. 1 Introduction
    1. 1.1 From one to multivariable renormalisation
    2. 1.2 Locality for multivariable meromorphic germs
    3. 1.3 Locality Galois groups and locality Lyndon words
    4. 1.4 Generalised evaluators and locality characters
    5. 1.5 Outline of the paper
  2. 2 Meromorphic germs with linear poles
    1. 2.1 Spaces of meromorphic germs with linear poles
    2. 2.2 Laurent expansions and the induced decompositions
    3. 2.3 Dependence subspaces
  3. 3 Locality transformation groups on meromorphic germs
    1. 3.1 Locality algebras of meromorphic germs
      1. 3.1.1 Locality algebras
      2. 3.1.2 Locality subalgebras of meromorphic germs
    2. 3.2 Automorphism groups of simplex locality algebras
    3. 3.3 Locality Galois groups
  4. 4 Locality polynomial algebras generated by Chen and Speer fractions
    1. 4.1 Locality shuffle algebras as locality polynomial algebras
    2. 4.2 Applications to ordered fractions
  5. 5 Locality characters and analytic renormalisation
    1. 5.1 Speer’s $s$-families and Speer fractions
    2. 5.2 Generalised evaluators on locality subalgebras of meromorphic germs
    3. 5.3 Locality Galois actions on generalised evaluators

##  1\. Introduction

This work seeks to reveal the rich structure of meromorphic germs in several
variables with linear poles, to describe subalgebras, to explore the structure
of transformation group, to evaluate them at poles in a consistent manner, and
to compare different evaluations. The locality framework developed by the
authors [CGPZ1], appears to be well suited to achieve these goals.

###  1.1. From one to multivariable renormalisation

Renormalisation is a procedure used to evaluate divergent expressions in
various areas of physics and mathematics, ranging from the classical instance
of Feynman amplitudes in perturbative quantum field theory [t’H, t’HV] to
multiple zeta functions at poles (see [GZ, MP] for example) and Todd functions
for toric varieties see, e.g. [BV2, P]. In either case, a preliminary step is
a regularisation procedure, after which one can extract divergences and
evaluate at the poles. There is a great freedom in the choice of
regularisation and the method of extracting divergences, specifically, a
regularisation can involve one or multiple parameters.

The algebraic structure underlying single parameter renormalisation has
attracted great interest in mathematics since the groundbreaking work of
Connes and Kreimer [CK1, CK2] to tackle Feynman integrals.

In their approach, the combinatorics of the divergent expressions are
organized into a connected Hopf algebra $\mathcal{H}$, while the
regularisations of the divergent expressions have their Laurent series
expansions in the Rota-Baxter algebra
${\mathcal{M}}(\mathbb{C})=\mathbb{C}[z^{-1},z]]$, characterised by its linear
decomposition

(1)
${\mathcal{M}}(\mathbb{C})={\mathcal{M}}_{-}(\mathbb{C})\oplus{\mathcal{M}}_{+}(\mathbb{C}),$

into subalgebras
${\mathcal{M}}_{-}(\mathbb{C}):=z^{-1}\mathbb{C}[z^{-1}],{\mathcal{M}}_{+}(\mathbb{C}):=\mathbb{C}[[z]]$
and with the induced projection

(2)
$\pi_{+}:{\mathcal{M}}(\mathbb{C})\longrightarrow{\mathcal{M}}_{+}(\mathbb{C}),\quad
f(z)=\sum_{k=-K}^{\infty}a_{k}\,z^{k}\mapsto\sum_{k=0}^{\infty}a_{k}\,z^{k}.$

The regularisation map is enriched to an algebra homomorphism

(3) $\phi:\mathcal{H}\to{\mathcal{M}}(\mathbb{C}),$

which then factorises according to the algebraic Birkhoff factorisation, as
the convolution product $\phi=\phi_{-}^{-1}\star\phi_{+}$ of a holomorphic
part $\phi_{+}$ with values in the subalgebra ${\mathcal{M}}_{+}(\mathbb{C})$
and a polar part $\phi_{-}$ with values in ${\mathcal{M}}_{-}(\mathbb{C})$.
The renormalised map $\phi_{+}$ which is then evaluated at the poles, is built
inductively with the recursion encoded in the coproduct, reflecting the
celebrated BPHZ procedure in perturbative quantum field theory [BP, Hep, Zi].

Extending the Connes-Kreimer approach to multiple parameter regularisations
leads to an algebra homomorphism

$\phi:\mathcal{H}\longrightarrow{\mathcal{M}}(\mathbb{C}^{\infty})$

with values in ${\mathcal{M}}(\mathbb{C}^{\infty})$, the algebra of
multivariable meromorphic germs at zero with linear poles for the Hopf algebra
of convex polyhedral cones [GPZ] and then later in a more general locality
framework [CGPZ1, CGPZ2, CGPZ3]. This approach uses in an essential way a
locality version of the Rota-Baxter algebraic structure on
${\mathcal{M}}(\mathbb{C}^{\infty})$.

###  1.2. Locality for multivariable meromorphic germs

In physics, the principle of locality is a key feature of field theory which
states that an object is influenced directly only by its immediate
surroundings. We interpret locality more generally as certain binary relations
[CGPZ1], enhance algebras to locality algebras and call locality morphisms,
the morphisms that preserve such locality relations. Its relation to causality
in quantum field theory was discussed in [Re]. In this paper the locality
relation on meromorphic germs is induced by an inner product $Q$ on the
underlying vector space in the following way. A pair of meromorphic germs lies
in the graph of the locality relation if the linear spaces spanned by the sets
of variables they respectively depend on, called their dependence subspaces,
are mutually orthogonal.This locality provides a natural splitting

${\mathcal{M}}(\mathbb{C}^{\infty})={\mathcal{M}}_{+}(\mathbb{C}^{\infty})\oplus{\mathcal{M}}_{-}^{Q}(\mathbb{C}^{\infty})$

into the subspace ${\mathcal{M}}_{+}(\mathbb{C}^{\infty})$ of holomorphic
germs and a space ${\mathcal{M}}_{-}^{Q}(\mathbb{C}^{\infty})$ of what we call
“polar germs”. While ${\mathcal{M}}_{-}^{Q}(\mathbb{C}^{\infty})$ is not a
subalgebra of ${\mathcal{M}}^{Q}(\mathbb{C}^{\infty})$, it bares the
remarkable property of being a locality ideal.

In applications [CGPZ2, CGPZ3], we equip the Hopf algebra with a locality
structure, turning it into a locality Hopf algebra, and $\phi$ becomes a
locality morphism of algebras. Thanks to the fact that
${\mathcal{M}}^{Q}(\mathbb{C}^{\infty})$ is a locality ideal, not only the
algebraic Birkhoff factorisation of $\phi$ can be recovered in the locality
setting, moreso, the renormalisation procedure $\phi_{+}$ simplifies to the
post composition $\pi_{+}^{Q}\circ\phi$ with the projection

$\pi_{+}^{Q}:{\mathcal{M}}(\mathbb{C}^{\infty})\to{\mathcal{M}}_{+}(\mathbb{C}^{\infty})$

on the holomorphic part. Its composition ${\rm ev}_{0}\circ\pi_{+}^{Q}$ with
the evaluation at zero ${\rm
ev}_{0}:{\mathcal{M}}_{+}(\mathbb{C}^{\infty})\to\mathbb{C}$ on the resulting
holomorphic germs, can be viewed as a multivariable minimal substraction
scheme.

A prototype of this approach was proposed in the pioneering work of Speer
[Sp2, Sp4] on analytic renormalisation in quantum field theory. This
multivariable renormalisation method was later implemented for the baby model
of Riemann integrals indexed by rooted trees in [CGPZ3] and further discussed
in [DZ].

Thus while passing from the classical approach of one parameter
renormalisation to the locality approach of multiple parameter
renormalisation, the focus of analysis is shifted from the source Hopf algebra
$\mathcal{H}$ of $\phi$ to the target algebra $\phi(\mathcal{H})$ in
${\mathcal{M}}(\mathbb{C}^{\infty})$ discussed in this paper.

###  1.3. Locality Galois groups and locality Lyndon words

In practice, we consider locality subalgebras
$\phi(\mathcal{H})\subset{\mathcal{A}}\subset{\mathcal{M}}(\mathbb{C}^{\infty})$.
A linear transformation $T$ on ${\mathcal{A}}$ induces another morphism

$T\circ\phi:\mathcal{H}\longrightarrow{\mathcal{A}}\subset{\mathcal{M}}(\mathbb{C}^{\infty}),$

from which we can again build a map

$\pi_{+}^{Q}\circ
T\circ\phi:\mathcal{H}\longrightarrow{\mathcal{M}}_{+}(\mathbb{C}^{\infty}).$

Locality isomorphisms of locality subalgebras
${\mathcal{A}}\subset{\mathcal{M}}(\mathbb{C}^{\infty})$, which restrict to
the identity map on ${\mathcal{M}}_{+}(\mathbb{C}^{\infty})$ form a group
${\rm Gal}^{Q}({\mathcal{A}}/{\mathcal{M}}_{+})$ we call locality Galois
groups. It plays the role of a renormalisation group [CK2, CM] in relating
different renormalisations.

Evaluating at zero by the map ${\rm ev}_{0}$, gives rise to a locality
character

${\mathcal{E}}_{T}:={\rm ev}_{0}\circ\pi_{+}^{Q}\circ
T\circ\phi:\mathcal{H}\longrightarrow\mathbb{C},$

which after Speer, we call an evaluator, depending on the choice of $T$.

To describe the action of the locality Galois group, we give a careful study
of the structure of locality subalgebras of
${\mathcal{M}}(\mathbb{C}^{\infty})$, defined by prescribed types of linear
poles. We focus on Chen type poles which typically arise from multiple zeta
functions and on the more general class of Speer fractions which arise in
Feynman integrals. We show that they both form locality polynomial algebras,
with a locality polynomial basis given a locality version of Lyndon words, we
call locality Lyndon words. For this purpose, we enhance to the locality
setup, the realisation of shuffle product algebras as polynomial algebras on
Lyndon words in the classical work of Chen-Fox-Lyndon and Radford [CFL, Ra].
As an application, transitivity is established for the action of the locality
Galois groups on the generalised evaluators on these two classes of locality
subalgebras.

###  1.4. Generalised evaluators and locality characters

In his seminal work [Sp1, Sp2, Sp3, Sp4] on analytic renormalisation, Speer
gives an axiomatic formulation for the evaluation of regularised quantities,
called generalised evaluators which he applies to spaces of meromorphic germs
in several variables singled out by the regularisation step in his study of
Feynman integrals. In the final renormalisation step, he proposes a
generalised evaluator defined by averaging over iterated one dimensional
evaluators successively applied in each variable.

Speer’s pioneering multivariable approach nevertheless lacks a covariance
property since it is coordinate dependent. To circumvent this coordinate
dependence, we require the evaluator to be multiplicative on products of germs
depending on variables which span mutually perpendicular spaces (the
dependence subspaces mentioned above) instead of them having disjoint sets of
variables as in Speer’s work. This paper provides a covariant counterpart of
Speer’s approach in a sound mathematical framework, with the aim of setting up
a general framework to tackle divergences in various contexts and with the
following three goals in mind.

######  Problem 1.1.

Equip the space of germs arising in Speer’s and other multiparameter
regularisations with appropriate locality polynomial algebra structures;

######  Problem 1.2.

Interpret Speer’s generalised evaluators as locality characters on the
corresponding (locality) algebras, leading to a general concept of locality
generalised evaluator;

######  Problem 1.3.

Build a transformation group which relates different locality evaluators.

###  1.5. Outline of the paper

In the coalgebraic approach to renormalisation in one variable à la Connes and
Kreimer, one calls upon an inductive procedure to deal with mutual
compensations of divergences among different levels. Instead, here we want to
avoid the occurence of such compensations by means of a multiparameter
regularisation, which enables us to regularise each subdivergence in an
autonomous way by introducing a different parameter at each level as we go
deeper in the subdivergences. This is possible using locality structures,
which take care of keeping the different levels separate in requiring the
regularisation map to be a locality algebra homomorphism with range in
${\mathcal{M}}_{\mathbb{Q}}$.

We first provide some background in Section 2 on the space
${\mathcal{M}}_{\mathbb{Q}}$ of meromorphic germs with linear poles and
rational coefficients on the filtered lattice space
$(\mathbb{R}^{\infty},\mathbb{Z}^{\infty})$. For a given inner product $Q$ on
the underlying space, a complement of the holomorphic germs
${\mathcal{M}}_{\mathbb{Q}+}$ is given by the subspace
${\mathcal{M}}^{Q}_{\mathbb{Q}-}$ of polar germs defined in Eq. (5). This
gives rise to Laurent expansions and various decompositions and invariants
(residues) in ${\mathcal{M}}_{\mathbb{Q}}$ (Theorem 2.5), which serve as the
building blocks of our further study. We further give a detailed study of
dependence spaces of meromorphic germs and their decompositions in §2.3
(Theorem 2.13), in order to define the orthogonality of meromorphic germs.

With the above orthogonality of meromorphic germs at hand, we carry out a
careful study of locality algebras in Section 3, focusing on subalgebras of
${\mathcal{M}}_{\mathbb{Q}}$. §3.1 gives a description of the structures of a
locality subalgebra
${\mathcal{M}}_{\mathbb{Q}+}^{Q}(\Pi^{Q}(\widehat{{\mathcal{S}}}))$ of
${\mathcal{M}}_{\mathbb{Q}}$, which contains ${\mathcal{M}}_{\mathbb{Q}+}$ and
is generated by a set of fractions ${\mathcal{S}}$. The automorphisms of such
a locality subalgebra, which fix the holomorphic germs and preserve residue
type invariants of polar germs, are shown to form a group in §3.2 and §3.3.
Inspired by Cartier’s cosmic Galois group [B, C, CM], we call it the locality
Galois group of the locality subalgebra (Definition 3.13). A reduction theorem
(Theorem 3.15) of locality Galois groups is obtained, which shows that there
is a subgroup of the locality Galois group that can be described by special
automorphisms of the locality subalgebra generated over $\mathbb{Q}$ by the
same set of fractions ${\mathcal{S}}$.

To obtain the structure of locality Galois groups, in Section 4 we first give
a locality variant of polynomial algebras (Definition 4.1) and then extend the
polynomial generation of shuffle product algebras by Lyndon words to the
locality setting (Theorem 4.4). We finally show (Theorem 4.9) that the
locality subalgebras generated by certain classes of fractions are locality
polynomial algebras. These include Chen fractions arising in multiple zeta
functions, described in Example 2.7 and Speer fractions described in Example
4.10, named after Speer in acknowledgment of his work on analytic
renormalisation [Sp2, Sp3, Sp4] (see also [BR, DZ]).

Finally in Section 5, we apply the developed results to revisit Speer’s
approach in the locality framework. We first show (Proposition 5.1) that the
space spanned by the fractions arising from Speer’s s-families is precisely
the space of the aforementioned Speer fractions, and hence it is a locality
polynomial algebra. This addresses Problem 1.1.

As an analog of Speer’s generalised evaluators, in §5.2 we introduce the
notion of locality generalised evaluators (Definition 5.2) on a locality
subalgebra of ${\mathcal{M}}_{{\mathbb{Q}}}$. They are linear forms
${\mathcal{E}}$ extending the usual evaluation ${\rm ev}_{0}$ at $0$ defined
on holomorphic germs and, in accordance with the locality principle, they are
required to obey the following locality multiplicativity:

$f_{1}\perp^{Q}f_{2}\Rightarrow{\mathcal{E}}(f_{1}\,f_{2})={\mathcal{E}}(f_{1})\,{\mathcal{E}}(f_{2}).$

A proptotype is the minimal subtraction evaluator ${{\mathcal{E}}^{Q}_{\rm
MS}}\coloneqq{\rm ev}_{0}\circ\pi_{+}^{Q}$, see Eq. (48).
We further show in Proposition 5.6, that our locality generalised evaluators
satisfy the conditions required by Speer for generalised evaluators modulo a
topological requirement (which lies out of the scope of this paper, and is
discussed in [DPS]) and when adapting the locality relation appropriately.
This addresses Problem 1.2. Along the way, we discuss the difference between
Speer’s coordinate dependent generalised evaluator
${\mathcal{E}}_{\bullet}^{\rm iter}$ and our covariant minimal subtraction
scheme ${\mathcal{E}}^{Q}_{\rm MS}$.

The locality Galois group naturally acts on locality generalised evaluators by
composition (Eq. (50)). When the locality subalgebra is generated by a
locality polynomial algebra of fractions, the action is shown to be transitive
(Theorem 5.7). Consequently, every locality generalised evaluator on such a
locality polynomial algebra factors through the minimal subtraction evaluator
${\mathcal{E}}^{Q}_{\rm MS}$. Applying these results to Chen fractions and
Speer fractions, we obtain Corollary 5.8, addressing Problem 1.3. Finally we
show that multiple zeta values naturally give rise to a locality generalised
evaluator ${\mathcal{E}}^{\zeta}$ on the locality polynomial algebra
${\mathcal{A}}^{\rm
Chen}:={\mathcal{M}}_{\mathbb{Q}+}^{Q}(\Pi^{Q}(\widehat{{\mathcal{F}}}^{\rm
Ch}))$ of meromorphic germs at zero with Chen type poles (Eq. (53)). Comparing
${\mathcal{E}}^{\zeta}$ with the minimal subtraction evaluator
${\mathcal{E}}^{Q}_{\rm MS}$ gives a natural element in the locality Galois
group.

##  2\. Meromorphic germs with linear poles

This section first summarises definitions and results of [GPZ3] on the space
of meromorphic germs with linear poles, and then studies the dependence space
of meromorphic germs and their decompositions.

###  2.1. Spaces of meromorphic germs with linear poles

We work in a filtered lattice Euclidean space
$(\mathbb{R}^{\infty},\mathbb{Z}^{\infty},Q)$, consisting of

  1. (i)

a filtered lattice space $(\mathbb{R}^{\infty},\mathbb{Z}^{\infty})$ defined
by direct limits

$\mathbb{R}^{\infty}\coloneqq\varinjlim\mathbb{R}^{k},\quad\mathbb{Z}^{\infty}\coloneqq\varinjlim\mathbb{Z}^{k},$

under the standard embeddings $i_{k}:\mathbb{R}^{k}\to\mathbb{R}^{k+1}$,

  2. (ii)

an inner product $Q$ on the filtered lattice space, defined by a family
$Q=(Q_{k})_{k\geq 1}$ of inner products

$Q_{k}:\mathbb{R}^{k}\otimes\mathbb{R}^{k}\to{\mathbb{R}},$

such that

$Q_{k+1}|_{\mathbb{R}^{k}\times\mathbb{R}^{k}}=Q_{k},\quad
Q_{k}(\mathbb{Z}^{k}\otimes\mathbb{Z}^{k})\subset\mathbb{Q}.$

For a field $\mathbb{K}$ with $\mathbb{Q}\subset\mathbb{K}\subset\mathbb{R}$,
we denote by
${\mathcal{L}}_{\mathbb{K}}(\mathbb{C}^{k})={\mathcal{L}}_{\mathbb{K}}(\mathbb{K}^{k}\otimes\mathbb{C})$
the space of linear forms on $\mathbb{C}^{k}$ which take $\mathbb{K}$-values
on $\mathbb{K}^{k}$.

On the filtered lattice space $(\mathbb{R}^{\infty},\mathbb{Z}^{\infty})$, a
meromorphic germ $f$ at zero on $\mathbb{R}^{k}\otimes\mathbb{C}$ is said to
be $\mathbb{K}$-holomorphic if it is a holomorphic germ at zero whose power
series expansion for any dual basis of $\mathbb{Z}^{k}$ has coefficients in
$\mathbb{K}$, and to have $\mathbb{K}$-linear poles if there are vectors
$L_{1},\ldots,L_{k}\in(\mathbb{Z}^{k})^{*}\otimes\mathbb{K}$ (possibly with
repetitions) such that

$f\,\Pi_{i=1}^{k}L_{i}$

is a $\mathbb{K}$-holomorphic germ.

######  Remark 2.1.

For the rest of the paper, all meromorphic germs are taken to be at zero
unless otherwise stated.

Let
${\mathcal{M}}_{\mathbb{K}}(\mathbb{C}^{k})={\mathcal{M}}_{\mathbb{K}}({\mathbb{R}}^{k}\otimes\mathbb{C})$
(resp.
${\mathcal{M}}_{\mathbb{K}+}(\mathbb{C}^{k})={\mathcal{M}}_{\mathbb{K}+}({\mathbb{R}}^{k}\otimes\mathbb{C})$)
denote the space of meromorphic (resp. $\mathbb{K}$-holomorphic) germs with
$\mathbb{K}$-linear poles.

The inner product $Q$ induces a family of linear bijections

$Q_{k}:\mathbb{K}^{k}\to(\mathbb{K}^{k})^{*},\quad u\mapsto Q_{k}(u,\cdot)$

and $Q_{k}^{-1}:(\mathbb{K}^{k})^{*}\to\mathbb{K}^{k}.$ This gives rise to
maps

$p_{k}\coloneqq Q_{k}^{-1}i_{k}^{*}Q_{k+1}:\mathbb{C}^{k+1}\to\mathbb{C}^{k}.$

Explicitly, let $W_{k}$ be the orthogonal complement of $\mathbb{C}^{k}$ in
$\mathbb{C}^{k+1}$. Then the map $p_{k}$ is the projection of
$\mathbb{C}^{k+1}$ to $\mathbb{C}^{k}$ along $W_{k}$ and give rise to the
directed systems:

$p_{k}:{\mathcal{M}}_{\mathbb{K}}(\mathbb{C}^{k})\to{\mathcal{M}}_{\mathbb{K}}(\mathbb{C}^{k+1}),\quad
p_{k}:{\mathcal{M}}_{\mathbb{K}+}(\mathbb{C}^{k})\to{\mathcal{M}}_{\mathbb{K}+}(\mathbb{C}^{k+1}),$

and the direct limits

${\mathcal{M}}_{\mathbb{K}}:={\mathcal{M}}_{\mathbb{K}}(\mathbb{C}^{\infty}):=\varinjlim{\mathcal{M}}_{\mathbb{K}}(\mathbb{C}^{k}),\quad{\mathcal{M}}_{\mathbb{K}+}:={\mathcal{M}}_{\mathbb{K}+}(\mathbb{C}^{\infty}):=\varinjlim{\mathcal{M}}_{\mathbb{K}+}(\mathbb{C}^{k})$

of spaces of meromorphic germs with $\mathbb{K}$-linear poles (resp.
$\mathbb{K}$-holomorphic germs).

By restriction, we also let

(4)
${\mathcal{L}}_{\mathbb{K}}\coloneqq{\mathcal{L}}_{\mathbb{K}}(\mathbb{C}^{\infty}):=\varinjlim{\mathcal{L}}_{\mathbb{K}}(\mathbb{C}^{k})$

be the direct limit of spaces of $\mathbb{K}$-linear forms. Notice that $Q$
induces an inner product in ${\mathcal{L}}_{\mathbb{K}}(\mathbb{C}^{\infty})$
which we still denote by $Q$.

As in [GPZ3], on a filtered lattice Euclidean space
$(\mathbb{R}^{\infty},\mathbb{Z}^{\infty},Q)$, we define a polar germ in
$\mathbb{C}^{k}$ with $\mathbb{K}$-coefficients to be a germ of meromorphic
functions of the form

(5) $\frac{h(\ell_{1},\ldots,\ell_{m})}{L_{1}^{s_{1}}\cdots
L_{n}^{s_{n}}},\vspace{-.3cm}$

where

  1. (i)

$h$ lies in ${\mathcal{M}}_{\mathbb{K}+}(\mathbb{C}^{m})$,

  2. (ii)

$\ell_{1},\ldots,\ell_{m},L_{1},\ldots,L_{n}$ lie in
${\mathcal{L}}_{\mathbb{K}}(\mathbb{C}^{k})$, with $L_{1},\ldots,L_{n}$
linearly independent, such that

$Q(\ell_{i},L_{j})=0\quad\forall i\in\\{1,\ldots,m\\},j\in\\{1,\ldots,n\\},$

  3. (iii)

$m$ is a nonnegative integer and $n,s_{1},\ldots,s_{n}$ are positive integers.

The convex cone (resp. subspace space) spanned by $L_{1},\ldots,L_{n}$ is
called the supporting cone (resp. supporting space) of the polar germ $f$. The
supporting space, independent of the presentation of the germ in the form of
the fraction (see [GPZ3, Lemma 2.9]), is denoted by ${\rm Supp}(f)$. Recall
from [GPZ3, Definition 5.1] that, for a polar germ
$\frac{h(\ell_{1},\ldots,\ell_{m})}{L_{1}^{s_{1}}\cdots L_{n}^{s_{n}}}$, the
integer

$\text{p-ord}\left(\frac{h(\ell_{1},\ldots,\ell_{m})}{L_{1}^{s_{1}}\cdots
L_{n}^{s_{n}}}\right)\coloneqq|(s_{1},\ldots,s_{n})|\coloneqq
s_{1}+\cdots+s_{n}\vspace{-.1cm}$

is well defined, called the p-order of the polar germ.

Let ${\mathcal{M}}^{Q}_{\mathbb{K}-}(\mathbb{C}^{k})$ denote the linear space
spanned by polar germs on $\mathbb{C}^{k}$ with $\mathbb{K}$-linear poles.
Again we have a directed system

$p_{k}:{\mathcal{M}}^{Q}_{\mathbb{K}-}(\mathbb{C}^{k})\to{\mathcal{M}}^{Q}_{\mathbb{K}-}(\mathbb{C}^{k+1})\vspace{-.2cm}$

and the direct limit

${\mathcal{M}}^{Q}_{\mathbb{K}-}\coloneqq{\mathcal{M}}^{Q}_{\mathbb{K}-}(\mathbb{C}^{\infty})\coloneqq\varinjlim{\mathcal{M}}^{Q}_{\mathbb{K}-}(\mathbb{C}^{k})\subseteq{\mathcal{M}}_{\mathbb{K}}(\mathbb{C}^{\infty})\vspace{-.1cm}$

of polar germs with $\mathbb{K}$-linear poles.

Polar germs split according to their supporting subspaces and p-orders as
follows.

######  Lemma 2.2.

[GPZ3, Lemma 3.5] Suppose $\sum\limits_{i=1}^{r}S_{i}=0$ for a sum of
holomorphic germs and $\mathbb{K}$-polar germs. For any linear
$\mathbb{K}$-subspace $W$ of $V$ and $N\in{\mathbb{Z}}_{>0}$, we have
$\sum\limits_{i}\\!^{\prime}S_{i}=0,$ where the sum is over the terms $S_{i}$
with ${\rm Supp}(S_{i})=W$ and $\text{\rm p-ord}(S_{i})=N$, with the
convention that the sum over an empty set is zero.

Following [GPZ3, Definition 3.2],

  1. (i)

a family of (convex) cones is called properly positioned if every pair of
cones in the family intersect along their faces, including the zero
dimensional face at $0$, and their union does not contain a straight line;

  2. (ii)

a family of polar germs properly positioned if, for each of the polar germs,
there is a choice of a supporting cone such that the resulting family of cones
is properly positioned;

  3. (iii)

a family of polar germs is called projectively properly positioned if it is
properly positioned and none of the denominators of the polar germs is
proportional to another.

Here is a useful criterion for the linear independence of polar germs.

######  Proposition 2.3.

[GPZ3, Proposition 3.6] A finite family of polar germs with projectively
properly positioned supporting cones is linearly independent.

###  2.2. Laurent expansions and the induced decompositions

There are several decompositions of meromorphic germs with linear poles.
Recall that a convex cone is called simplicial if it is spanned by a set of
linearly independent vectors.

######  Proposition 2.4.

[GPZ3, Theorem 4.13.] For any $f$ in ${\mathcal{M}}_{\mathbb{K}}$, there exist
a properly positioned family ${\underline{{C}}}$ of simplicial cones together
with a family of $\mathbb{K}$-polar germs $\\{S_{j}\\}_{j\in J}$ supported on
${\underline{{C}}}$ $($in the sense that a supporting cone of each $S_{j}$ is
in ${\underline{{C}}})$, and a holomorphic germ $h$, such that

(6) $f=\sum_{j\in J}S_{j}+h.$

Eq. (6) is called a Laurent expansion of $f$ supported on ${\underline{{C}}}$
and it is unique up to subdivisions of the properly positioned family of
simplicial cones.

For $p\in\mathbb{Z}_{\geq 0}$, $d\in\mathbb{Z}_{\geq 0}$ and a finite
dimensional $\mathbb{K}$-subspace $U\subset\mathbb{R}^{\infty}$, let

  * •

${\mathcal{M}}_{\mathbb{K}}^{p}$ denote the linear span of $\mathbb{K}$-polar
germs of p-order $p$;

  * •

${\mathcal{M}}_{\mathbb{K},d}$ denote the linear span of $\mathbb{K}$-polar
germs whose supporting cones have dimension $d$;

  * •

${\mathcal{M}}_{\mathbb{K},U}$ denote the linear span of $\mathbb{K}$-polar
germs with supporting space $U$.

######  Theorem 2.5.

[GPZ3, Theorem 5.3] We have the decompositions

(7) $\displaystyle{{\mathcal{M}}}_{\mathbb{K}}$ $\displaystyle=$
$\displaystyle\bigoplus_{p\geq 0}{{\mathcal{M}}}_{\mathbb{K}}^{p},$ (8)
$\displaystyle{{\mathcal{M}}}_{\mathbb{K}}$ $\displaystyle=$
$\displaystyle\bigoplus_{d\geq 0}{{\mathcal{M}}}_{\mathbb{K},d},$ (9)
$\displaystyle{{\mathcal{M}}}_{\mathbb{K}}$ $\displaystyle=$
$\displaystyle\bigoplus_{U\subset\mathbb{R}^{\infty}}{{\mathcal{M}}}_{\mathbb{K},U}.$

In particular, there is a decomposition $($see also [BV1]$)$:

${\mathcal{M}}_{\mathbb{K}}={\mathcal{M}}_{\mathbb{K}+}\oplus{\mathcal{M}}^{Q}_{\mathbb{K}-}.$

We give further notations.

  1. (i)

Corresponding to the decomposition in Eq. (7), let $q$ be the highest p-order
of the polar germs in a (thus every) Laurent expansion of $f$. Define the
p-residue of $f$ [GPZ3, Definition 6.1] by

(10) ${\rm pRes}(f)\,{:}=\sum_{\text{\rm
p-ord}(S_{i})=q}\frac{h_{i}(0)}{\vec{L}_{i}^{\vec{s}_{i}}}.$

  2. (ii)

Corresponding to the decomposition in Eq. (8), let $e$ be the largest among
the dimensions of the supporting spaces of the polar germs in a (thus every)
Laurent expansion of $f$. Define the d-residue of $f$ by

(11) ${\rm dRes}(f)\,{:}=\sum_{{\rm dim}({\rm
Supp}(S_{i}))=e}\frac{h_{i}(0)}{\vec{L}_{i}^{\vec{s}_{i}}}.$

######  Remark 2.6.

As proved in [GPZ3, Proposition 6.2], the p-residue of a meromorphic germ with
linear poles depends neither on the choice of a Laurent expansion nor on the
choice of the inner product. The d-residue does not depend on the choice of a
Laurent expansion, but it does depend on the choice of the inner product.

###  2.3. Dependence subspaces

For any subset $U$ of ${\mathcal{M}}_{\mathbb{Q}}$, let $\mathbb{Q}U$ denote
the $\mathbb{Q}$-subspace of ${\mathcal{M}}_{\mathbb{K}}$ spanned by $U$.

A simplex fraction is a fraction of the form $\frac{1}{L_{1}^{s_{1}}\cdots
L_{k}^{s_{k}}}$, where $L_{1},\ldots,L_{k}\in{\mathcal{L}}_{\mathbb{Q}}$ are
linearly independent and $s_{i}\in\mathbb{Z}_{>0}$, $i=1,\ldots,k$. Let
${\mathcal{F}}$ be the set of all simplex fractions over $\mathbb{Q}$. Then
for any inner product $Q$ in $(\mathbb{R}^{\infty},\mathbb{Z}^{\infty})$, we
trivially have
$\mathbb{Q}{\mathcal{F}}\subset{\mathcal{M}}_{\mathbb{Q}-}^{Q}.$

######  Example 2.7.

In the Euclidean filtered lattice space
$(\mathbb{R}^{\infty},\mathbb{Z}^{\infty},Q)$, let
$\mathcal{B}\coloneqq(e_{i})_{i\in\mathbb{Z}_{>0}}$ be an orthonormal basis.
Let $z_{i}$ be the coordinate function corresponding to $e_{i}$. A fraction of
the form

(12)
${\mathfrak{f}}\Big{(}\begin{array}[]{c}\scriptstyle{s_{1},\ldots,s_{k}}\\\
\scriptstyle{{u_{1}},\ldots,{u_{k}}}\end{array}\Big{)}\coloneqq\frac{1}{z_{u_{1}}^{s_{1}}(z_{u_{1}}+z_{u_{2}})^{s_{2}}\cdots(z_{u_{1}}+z_{u_{2}}+\cdots+z_{u_{k}})^{s_{k}}}\,,\
u_{i},s_{i}\in\mathbb{Z}_{>0},k\in\mathbb{N},u_{i}\neq u_{j}\,{\rm if}\ i\neq
j,$

is called a Chen fraction. The set of Chen fractions is denoted by

${\mathcal{F}}^{\rm Ch}\coloneqq{\mathcal{F}}^{{\rm Ch},Q,\mathcal{B}}.$

We borrow the following definitions from [GPZ3]. A meromorphic function $f$ on
$\mathbb{C}^{k}$ of the form $f=g(L_{1},\ldots,L_{n})$, where
$L_{1},\ldots,L_{n}$ are linear forms on $\mathbb{C}^{k}$ and $g$ a
meromorphic function on $\mathbb{C}^{n}$, is said to depend on the linear
subspace of $(\mathbb{C}^{k})^{*}$ spanned by $L_{1},\ldots,L_{n}$. One can
check that if $f$ depends on $V_{1}$ and $V_{2}$, then it depends on
$V_{1}\cap V_{2}$. Thus it makes sense to set the following definition.

######  Definition 2.8.

The dependence subspace ${\rm Dep}(f)$ of $f$ is the smallest linear subspace
of $(\mathbb{C}^{k})^{*}$ on which $f$ depends.

######  Example 2.9.

Let $(e_{1},e_{2},\ldots)$ be an orthonormal basis of
$(\mathbb{R}^{\infty},\mathbb{Z}^{\infty},Q)$.

${\rm
Dep}\left(\frac{1}{z_{1}(z_{1}+z_{2})}+\frac{1}{z_{2}(z_{1}+z_{2})}-\frac{2}{z_{1}(z_{1}+2z_{2})}-\frac{1}{z_{2}(z_{1}+2z_{2})}+\frac{1}{z_{3}}\right)=\\{e_{3}\\},$

since the sum of the first four terms is zero.

######  Definition 2.10.

Two meromorphic germs $f$ and $g$ in ${\mathcal{M}}_{\mathbb{Q}}$ are called
$Q$-orthogonal, which we write $f\perp^{Q}g$, if their dependence subspaces
are orthogonal.

######  Example 2.11.

  1. (i)

Let $(e_{1},e_{2},\ldots)$ be an orthonormal basis of
$(\mathbb{R}^{\infty},\mathbb{Z}^{\infty},Q)$. We have

$\frac{1}{z_{1}+z_{2}}\,\perp^{Q}\,(z_{1}-z_{2}).\vspace{-.1cm}$

  2. (ii)

Polar germs are precisely germs of the form $h/M$ for $h$ in
${\mathcal{M}}_{\mathbb{Q}+}(\mathbb{C}^{\infty})$ and $M$ given by products
of powers of linearly independent linear forms, such that $h\perp^{Q}M$.

An element $f\in{\mathcal{M}}_{\mathbb{Q}}$ is of the form
$f=\frac{h}{\ell_{1}\cdots\ell_{r}}$ for a holomorphic germ $h$ and linear
forms $\ell_{1},\ldots,\ell_{r}$. The next lemma shows that the factors in the
fraction can be chosen to have their dependence subspaces contained in the
dependence subspace of $f$.

######  Lemma 2.12.

For any $f$ in ${\mathcal{M}}_{\mathbb{Q}}$, there are linear forms
$\ell_{i}=\ell_{i}(L_{1},\ldots,L_{n}),i=1,\ldots,p,$ and a holomorphic germ
$h=h(L_{1},\ldots,L_{n})$ for a basis $L_{1},\ldots,L_{n}$ of ${\rm Dep}(f)$,
such that $\displaystyle{f=\frac{h}{\ell_{1}\cdots\ell_{p}}}.$

###### Proof.

Let $f$ be in ${\mathcal{M}}_{\mathbb{Q}}(\mathbb{C}^{k})$ for some $k\geq 1$.
We extend a basis $L_{1},\ldots,L_{n}$ of ${\rm Dep}(f)$ to a basis
$L_{1},\ldots,L_{n},\ldots,L_{k}$ of $(\mathbb{C}^{k})^{*}$. Since $f$ is in
${\mathcal{M}}_{\mathbb{Q}}(\mathbb{C}^{k})$, there are linear combinations
$\ell_{1},\ldots,\ell_{m}$ of $L_{1},\ldots,L_{k}$ such that the product
$\ell_{1}\cdots\ell_{m}\,f$ is in
${\mathcal{M}}_{\mathbb{Q}+}(\mathbb{C}^{k}),$ that is,

(13)
$\ell_{1}\cdots\ell_{m}\,f(L_{1},\ldots,L_{n})\in{\mathcal{M}}_{\mathbb{Q}+}(\mathbb{C}^{k}).$

By rearrangement, we can assume that $\ell_{1},\ldots,\ell_{p}$ are linear
combinations of $L_{1},\ldots,L_{n}$ only; while $\ell_{p+1},\ldots,\ell_{m}$
have nontrivial linear contributions from the extra linear forms
$L_{n+1},\ldots,L_{k}$. Then we can choose a tuple
$(a_{n+1},\ldots,a_{k})\in\mathbb{C}^{k-n}$ such that the maps

$(L_{1},\ldots,L_{n})\mapsto\lambda_{i}(L_{1},\ldots,L_{n})\coloneqq\ell_{i}(L_{1},\ldots,L_{n},a_{n+1},\ldots,a_{k}),\quad
i=p+1,\ldots,m,$

are affine with $\lambda_{i}(0,\ldots,0)\neq 0$. Consequently, the maps
$(L_{1},\ldots,L_{n})\longmapsto\frac{1}{\lambda_{i}(L_{1},\ldots,L_{n})}$ are
holomorphic germs.

Thus setting $L_{n+1}=a_{n+1},\ldots,L_{k}=a_{k}$ in Eq. (13) yields a
holomorphic germ

$h:(L_{1},\ldots,L_{n})\longmapsto\ell_{1}\cdots\ell_{p}\,\lambda_{p+1}\cdots\lambda_{m}\,f,$

from which we define another holomorphic germ

$\tilde{h}(L_{1},\ldots,L_{n})\coloneqq\frac{h(L_{1},\ldots,L_{n})}{\lambda_{p+1}\cdots\lambda_{m}}.\vspace{-.2cm}$

Hence,

$f=f(L_{1},\ldots,L_{n})=\frac{h(L_{1},\ldots,L_{n})}{\ell_{1}\cdots\ell_{p}\lambda_{p+1}\cdots\lambda_{m}}=\frac{\tilde{h}(L_{1},\ldots,L_{n})}{\ell_{1}(L_{1},\ldots,L_{n})\cdots\ell_{p}(L_{1},\ldots,L_{n})}\vspace{-.1cm}$

is of the desired form. ∎

######  Theorem 2.13.

Write a germ $f$ in ${\mathcal{M}}_{\mathbb{Q}}$ according to the
decomposition in Eq. (9)$:$

(14) $f=\sum_{U\in\mathcal{U}}f_{U}+f_{0},$

where $\mathcal{U}$ is a finite set of nonzero finite-dimensional subspaces of
${\mathbb{R}}^{\infty}$, $0\neq f_{U}$ is a sum of polar germs with supporting
space $U$ and $f_{0}$ lies in ${\mathcal{M}}_{\mathbb{Q}+}$. We have

${\rm Dep}(f)=\sum_{U\in\mathcal{U}}{\rm Dep}(f_{U})+{\rm Dep}(f_{0}).$

###### Proof.

Since $f=\sum f_{U}+f_{0}$, clearly we have

${\rm Dep}(f)\subset\sum_{U\in\mathcal{U}}{\rm Dep}(f_{U})+{\rm Dep}(f_{0}).$

It remains to show that ${\rm Dep}(f_{U})\subset{\rm Dep}(f)$ for all
$U\in\mathcal{U}$ and ${\rm Dep}(f_{0})\subset{\rm Dep}(f)$.

By Lemma 2.12, there are linear forms $\ell_{1},\ldots,\ell_{p}$ and a
homomorphic germ $h$, all with dependent spaces in ${\rm Dep}(f)$ such that

$f=\frac{h}{\ell_{1}\cdots\ell_{p}}=\frac{h(L_{1},\ldots,L_{n})}{\ell_{1}(L_{1},\ldots,L_{n})\cdots\ell_{p}(L_{1},\ldots,L_{n})},$

for a basis $L_{1},\ldots,L_{n}$ of ${\rm Dep}(f)$. Then we can take the
Laurent expansion of $f$ in ${\rm Dep}(f)$ by [GPZ3, Theorem 2.11]. Thus for
all the polar germs in this Laurent expansion of $f$, their linear poles and
holomorphic numerators have dependence space in ${\rm Dep}(f)$. This gives
another decomposition

$f=g_{0}+\sum_{V\subset{\rm Dep}(f)}g_{V}$

according to Eq. (9). Comparing with the decomposition of $f$ in Eq. (14) as a
sum of a holomorphic germ and polar germs, we have

$f=f_{0}+\sum_{U\subset\mathcal{U}}f_{U}=g_{0}+\sum_{V\subset{\rm
Dep}(f)}g_{V}.$

Using the uniqueness of the decomposition in Eq. (9), we infer that for any
subspace $U\subset\mathcal{U}$ (resp. $U=0$), there is a space $V\subset{\rm
Dep}(f)$ (resp. $V=0$) such that $f_{U}=g_{V}$. This implies that ${\rm
Dep}(f_{U})$ is contained in ${\rm Dep}(f)$. Thus the proof is completed. ∎

######  Lemma 2.14.

For a nonzero rational linear combination $f=\sum_{i\in
I}\alpha_{i}S_{i}\in{\mathcal{M}}_{\mathbb{Q}}(\mathbb{C}^{k})$ of simplex
fractions $S_{i},i\in I,$ with the same supporting space $U$, we have ${\rm
Dep}(f)=U.$

###### Proof.

Clearly, ${\rm Dep}(f)\subseteq U.$ Suppose ${\rm Dep}(f)\subsetneq U$.
Theorem 2.13 gives

$f=g_{0}+\sum_{V\subset{\rm Dep}(f)}g_{V}$

where the terms in $g_{V}$ have supporting space $V$. Hence

$0=f-\sum_{i\in I}\alpha_{i}\,S_{i}=g_{0}+\sum_{V\subset{\rm
Dep}(f)}g_{V}-\sum_{i\in I}\alpha_{i}\,S_{i}.$

From ${\rm Dep}(f)\subsetneq U$, we have $V\subsetneq U$ in the above sum.
Thus the above sum is the decomposition of $0$ according to the supporting
spaces in Eq. (9), in which $\sum_{i\in I}\alpha_{i}\,S_{i}$ is the component
with supporting space $U$. Thus $\sum_{i\in I}\alpha_{i}\,S_{i}=0$, meaning
$f=0$. This is a contradiction. ∎

This leads to the following statement on sums of polar germs.

######  Proposition 2.15.

If $f=\sum f_{i}$ is a nonzero sum of polar germs $f_{i}$ with the same
supporting space $U$, then $U$ is a subset of ${\rm Dep}(f).$

###### Proof.

This follows from Lemma 2.14 after evaluation of the numerators of the polar
germs $f_{i}$ at appropriate arguments in the spirit of the proof of [GPZ3,
Theorem 3.7]. Indeed, let us write the polar germs $f_{i}=h_{i}\,S_{i}$ where

$S_{i}=\frac{1}{L_{1}^{s_{1}}\cdots L_{n}^{s_{n_{i}}}},\quad
s_{j}\in\mathbb{Z}_{\geq 0},\vspace{-.2cm}$

are simplex fractions with the same supporting space and $h_{i}$ are
holomorphic germs in some common set of variables $\ell_{n+1},\ldots,\ell_{k}$
which complete the independent linear forms $L_{1},\ldots,L_{n}$ arising in
the $S_{i}$’s to an orthonormal basis of $\mathbb{R}^{k}$. Since $f\neq 0$ we
can assume without loss of generality that none of the holomorphic germs
$h_{i}$ is identically zero. Hence, there is some tuple
$(\ell_{n+1}^{0},\ldots,\ell_{k}^{0})$ such that $\alpha_{i}\coloneqq
h_{i}(\ell_{n+1}^{0},\ldots,\ell_{k}^{0})\neq 0$ for all $i$. We write
$f=f(L_{1},\ldots,L_{n},\ell_{n+1},\ldots,\ell_{k})$ and take the
specialisation $g(L_{1},\ldots,L_{n})\coloneqq
f(L_{1},\ldots,L_{n},\ell_{n+1}^{0},\ldots,\ell_{k}^{0})$. Then ${\rm
Dep}(f)\supset{\rm Dep}(g)$. Applying Lemma 2.14 to
$g=\sum_{i}h_{i}(\ell_{n+1}^{0},\ldots,\ell_{k}^{0})\,S_{i}$ with
$\alpha_{i}=h_{i}(\ell_{n+1}^{0},\ldots,\ell_{k}^{0})$, we obtain ${\rm
Dep}(g)={\rm Supp}(g)$. This completes the proof. ∎

The following result shows that without loss of generality, we can assume that
a sum $f$ of polar germs with the same supporting space can be written as a
sum of polar germs whose numerators are holomorphic germs with dependence
space in ${\rm Dep}(f)$.

######  Proposition 2.16.

Let $f=\sum h_{i}\,S_{i}$ be a nonzero sum of polar germs with the same
supporting space, where $S_{i}$ is a simplex fraction and $h_{i}$ is a
holomorphic germ. Then $f$ can be written as a sum
$f=\sum\tilde{h}_{i}\,S_{i}$ of polar germs where the $\tilde{h}_{i}$’s are
now holomorphic germs with dependence spaces in ${\rm Dep}(f)$.

###### Proof.

By Proposition 2.15, the common supporting space $U$ lies in ${\rm Dep}(f)$.
Let $L_{1},\ldots,L_{n}$ be a basis of $U$ which we extend to a basis
$L_{1},\ldots,L_{n},\ell_{n+1},\ldots,\ell_{m}$ of ${\rm Dep}(f)$ and then
further to a basis $L_{1},\ldots,L_{n},\ell_{n+1},\ldots,\ell_{k}$ of
$(\mathbb{C}^{k})^{*}$ with $Q(L_{i},\ell_{j})=0,1\leq i\leq n,n+1\leq j\leq
k$. Thus, $S_{i}$ is a simplex fraction in the variables $L_{1},\ldots,L_{n}$
and

$f=f(L_{1},\ldots,L_{n},\ell_{n+1},\ldots,\ell_{k})=\sum
h_{i}(\ell_{n+1},\ldots,\ell_{k})\,S_{i}(L_{1},\ldots,L_{n}).$

By the definition of dependence space, $f$ does not depends on
$\ell_{m+1},\ldots,\ell_{k}$, so

$f=f(L_{1},\ldots,L_{n},\ell_{n+1},\ldots,\ell_{m},0,\ldots,0)=\sum
h_{i}(\ell_{n+1},\ldots,\ell_{m},0,\ldots,0)\,S_{i}(L_{1},\ldots,L_{k})$

as announced. ∎

##  3\. Locality transformation groups on meromorphic germs

In this section, we study meromorphic germs with linear poles in the context
of locality algebras. We then introduce the locality Galois group defined as a
group of automorphisms of meromorphic germs in this locality framework.

###  3.1. Locality algebras of meromorphic germs

We give general background on locality algebras and then focus on locality
subalgebras of meromorphic germs.

####  3.1.1. Locality algebras

We recall notations on locality structures from [CGPZ1].

######  Definition 3.1.

A locality set is a couple $(X,\top)$ where $X$ is a set and

$\top\coloneqq X\times_{\top}X\subseteq X\times X$

is a binary symmetric relation, called a locality relation, on $X$. For
$x_{1},x_{2}\in X$, denote $x_{1}\top x_{2}$ if $(x_{1},x_{2})\in\top$.

For a subset $U\subset X$, the polar subset of $U$ is

$U^{\top}\coloneqq\\{x\in X\,|\,(x,U)\subseteq\top\\}.$

For locality sets $(X,\top_{X})$ and $(Y,\top_{Y})$, a map $f:X\to Y$ is
called a locality map if

(15) $x_{1}\top_{X}x_{2}\Longrightarrow f(x_{1})\top_{Y}f(x_{2}),\quad\forall
x_{1},x_{2}\in X.$

We give some examples that will be further explored in the sequel.

######  Example 3.2.

  1. (i)

For any nonempty set $X$, being distinct: $x_{1}\top x_{2}$ if $x_{1}\neq
x_{2},$ defines a locality relation on $X$;

  2. (ii)

The $Q$-orthogonality relation
$\perp^{Q}\subset{\mathcal{M}}_{\mathbb{Q}}\times{\mathcal{M}}_{\mathbb{Q}}$
of Definition 2.10, turns ${\mathcal{M}}_{\mathbb{Q}}$ into a locality set.

  3. (iii)

Let $(X,\top)=({\mathbb{Z}}_{>0},\top)$ be the locality set in (i) and
$({\mathcal{M}}_{\mathbb{Q}},\perp^{Q})$ the locality set in (ii). With the
notation in Example 2.7, the map

$f:X\to{\mathcal{M}}_{\mathbb{Q}},\quad
n\mapsto{\mathfrak{f}}\Big{(}\begin{array}[]{c}\scriptstyle{1}\\\
\scriptstyle{n}\end{array}\Big{)}\coloneqq\frac{1}{z_{n}},\quad n>1,$

is a locality map.

Other algebraic structures can be generalised to the locality setting.

######  Definition 3.3.

  1. (i)

A locality vector space is a vector space $V$ equipped with a locality
relation $\top$ which is compatible with the linear structure on $V$ in the
sense that, for any subset $X$ of $V$, $X^{\top}$ is a linear subspace of $V$.

  2. (ii)

A (nonunitary) locality algebra over $K$ is a locality vector space $(A,\top)$
over $K$ together with a map

$m_{A}:A\times_{\top}A\to A,(u,v)\mapsto u\cdot v=m_{A}(x,y)\quad\text{for all
}(u,v)\in A\times_{\top}A$

satisfying the following variations of the associativity and distributivity.

    1. (a)

For $u,v,w\in A$ with $u\top v,u\top w,v\top w$, we have

(16) $(u\cdot v)\top w,\quad u\top(v\cdot w),\quad(u\cdot v)\cdot
w=u\cdot(v\cdot w).$

    2. (b)

For $u,v,w\in A$ with $u\top w,v\top w$ (and hence $(u+v)\top w,w\top(u+v)$),
we have

$(u+v)\cdot w=u\cdot w+v\cdot w,\quad w\cdot(u+v)=w\cdot u+w\cdot v,$
$(ku)\cdot w=k(u\cdot w),\quad u\cdot(kw)=k(u\cdot w),\ k\in K.$

  3. (iii)

A unitary locality algebra is a locality algebra $(A,\top,m_{A})$ with a unit
$1_{A}$ such that, for each $u\in A$, we have $1_{A}\top u$ and

$1_{A}\cdot u=u\cdot 1_{A}=u.$

We shall omit explicitly mentioning the unit $1_{A}$ unless doing so generates
ambiguity.

  4. (iv)

Let $(A,\top_{A})$ be a locality algebra. A subspace $B$ of $A$ is called a
(resp. unitary) locality subalgebra of $A$ if, with the restricted relation

$\top_{B}\coloneqq\top_{A}\cap(B\times B)\vspace{-.2cm}$

of $\top_{A}$ to $B$, the pair $(B,\top_{B})$ is a (resp. unitary) locality
algebra.

  5. (v)

Let $(A,\top_{A})$ be a commutative locality algebra and $C$ a unitary
locality subalgebra of $A$. A subspace $B$ of $A$ is called a (resp. unitary)
locality $C$-subalgebra of $A$ if $B$ is a (resp. unitary) locality subalgebra
of $A$ that contains $C$.

Given two locality algebras $(A_{i},\top_{i}),i=1,2$, a (resp. unitary)
locality algebra homomorphism is a linear map $\varphi:A_{1}\longrightarrow
A_{2}$ such that $a\top_{1}b$ implies $\varphi(a)\top_{2}\varphi(b)$ and
$\varphi(a\cdot b)=\varphi(a)\cdot\varphi(b)$ (resp. and
$\varphi(1_{A_{1}})=1_{A_{2}}$).

######  Example 3.4.

[CGPZ1, Corollary 3.23] With the relation $\perp^{Q}$ of Definition 2.10, the
pair $({\mathcal{M}}_{\mathbb{Q}},\perp^{Q})$ is a unitary locality algebra
and the projection

(17)
$\pi_{+}^{Q}:{\mathcal{M}}_{\mathbb{Q}}={\mathcal{M}}_{\mathbb{Q}+}\oplus{\mathcal{M}}^{Q}_{\mathbb{Q}-}\to{\mathcal{M}}_{\mathbb{Q}+}\vspace{-.1cm}$

along ${\mathcal{M}}^{Q}_{\mathbb{Q}-}$ is a unitary locality algebra
homomorphism.

######  Definition 3.5.

For a unitary locality algebra $(A,\top)$, a unitary locality endomorphism of
$A$ is a locality automorphism if it is invertible, preserves the unit, and
the inverse map is a locality algebra homomorphism. Let ${\rm Aut}^{\top}(A)$
denote the set of locality automorphisms of $A$.

We note that ${\rm Aut}^{\top}(A)$ forms a group for the composition, called
the locality automorphism group of $A$. There are counter examples that a
bijective locality homomorphism needs not be a locality automorphism.

####  3.1.2. Locality subalgebras of meromorphic germs

In the sequel, we consider locality subalgebras $({\mathcal{A}},\perp^{Q})$ of
the locality algebra $({\mathcal{M}}_{\mathbb{Q}},\perp^{Q})$ in Example 3.4.
For a subset $U$ of ${\mathcal{M}}_{\mathbb{Q}}$, let

(18) $\widehat{U}\coloneqq U\cup\\{1\\},\vspace{-.2cm}$

with $1$ being the constant function.

We first give the structure of locality subalgebras of
${\mathcal{M}}_{\mathbb{Q}}$ generated by a set, with rational coefficients or
${\mathcal{M}}_{\mathbb{Q}+}$ coefficients. As we shall see, a careful
analysis using tools such as supporting and dependent spaces is needed when
extending the notion of subalgebra to the locality setting.

Given a subset $\mathcal{U}$ of ${\mathcal{M}}_{\mathbb{Q}}$, let

$\Pi^{Q}(\mathcal{U})\coloneqq\bigg{\\{}\prod_{i}s_{i}\,\bigg{|}\,s_{i}\in\mathcal{U},\forall
i,s_{i}\perp^{Q}s_{j},\,\forall i\not=j\ \bigg{\\}}\vspace{-.2cm}$

be the set of meromorphic germs locality generated by $\mathcal{U}$. With the
notation of Eq. (18), we have

$\Pi^{Q}(\widehat{\mathcal{U}})=\Pi^{Q}(\mathcal{U})\cup\\{1\\}.\vspace{-.2cm}$

######  Proposition 3.6.

Given a set ${\mathcal{S}}$ of simplex fractions, the subspace of
$\mathbb{Q}\widehat{{\mathcal{F}}}$

${\mathbb{Q}}\Pi^{Q}(\widehat{{\mathcal{S}}})\coloneqq\bigg{\\{}\sum_{i}c_{i}\,S_{i}\,\bigg{|}\,c_{i}\in\mathbb{Q},S_{i}\in\Pi^{Q}(\widehat{{\mathcal{S}}})\bigg{\\}}\vspace{-.2cm}$

spanned by $\Pi^{Q}(\widehat{{\mathcal{S}}})$ is a unitary locality subalgebra
of $\mathbb{Q}{\mathcal{F}}$.

Thus ${\mathbb{Q}}\Pi^{Q}(\widehat{{\mathcal{S}}})$ is the unitary locality
subalgebra of $\mathbb{Q}\widehat{{\mathcal{F}}}$ generated by
${\mathcal{S}}$.

###### Proof.

Since $c_{0}=1$ serves as the unit, we just need to prove that for $f,g$ in
$\Pi^{Q}({\mathcal{S}})$ with $f\perp^{Q}g$, $fg$ lies in
$\mathbb{Q}\Pi^{Q}({\mathcal{S}})$. According to the grading in Eq. (9), we
write

$f=\sum_{U}f_{U},\quad g=\sum_{V}g_{V},\vspace{-.1cm}$

where $f_{U}$ is the sum of simplex fractions with the same supporting space
$U$ and $g_{V}$ is the sum of simplex fractions with the same supporting space
$V$. By Theorem 2.13, we have

${\rm Dep}(f)=\sum_{U}{\rm Dep}(f_{U}),\quad{\rm Dep}(g)=\sum_{V}{\rm
Dep}(g_{V})\vspace{-.1cm}$

from which we infer that, for any $U$ and $V$ appearing in the decompositions
of $f$ and $g$,

$f\perp^{Q}g\Longrightarrow f_{U}\perp^{Q}g_{V}.$

By Lemma 2.14, each $f_{U}\not=0$ (resp. $g_{V}\not=0$), being a sum of simple
fractions with the same supporting space $U$ (resp. $V$), gives $U={\rm
Dep}(f_{U})$ (resp. $V={\rm Dep}(g_{V})$). Thus we have

$U\perp^{Q}V.$

Hence the products $f_{U}g_{V}$ arising in the decomposition
$f\,g=\sum_{U,V}f_{U}g_{V}$ lie in $\mathbb{Q}\Pi^{Q}({\mathcal{S}})$.
Consequently the product $f\,g$ also lies in
$\mathbb{Q}\Pi^{Q}({\mathcal{S}})$. ∎

######  Proposition 3.7.

Given a set ${\mathcal{S}}$ of simplex fractions, the set

${\mathcal{M}}_{\mathbb{Q}+}^{Q}\big{(}\Pi^{Q}(\widehat{{\mathcal{S}}})\big{)}\coloneqq\bigg{\\{}\sum_{i}h_{i}S_{i},\bigg{|}\,h_{i}\in{\mathcal{M}}_{\mathbb{Q}+},S_{i}\in\Pi^{Q}(\widehat{{\mathcal{S}}}),h_{i}\perp^{Q}S_{i}\bigg{\\}}\vspace{-.1cm}$

is a unitary locality subalgebra of ${\mathcal{M}}_{\mathbb{Q}}$.

Thus
${\mathcal{M}}_{\mathbb{Q}+}^{Q}\big{(}\Pi^{Q}(\widehat{{\mathcal{S}}})\big{)}$
is the unitary locality ${\mathcal{M}}_{\mathbb{Q}+}$-subalgebra of
${\mathcal{M}}_{\mathbb{Q}}$ generated by ${\mathcal{S}}$.

###### Proof.

Clearly, ${\mathcal{M}}_{\mathbb{Q}+}^{Q}(\Pi^{Q}(\widehat{{\mathcal{S}}}))$
is a vector space. As in the proof of Proposition 3.6, we only need to verify
that, for
$a,b\in{\mathcal{M}}_{\mathbb{Q}+}^{Q}(\Pi^{Q}(\widehat{{\mathcal{S}}}))$ with
$a\perp^{Q}b$, the locality product $ab$ is well defined and lies in
${\mathcal{M}}_{\mathbb{Q}+}^{Q}(\Pi^{Q}(\widehat{{\mathcal{S}}}))$. To
complete this, in accordance with the grading in Eq. (9), write

$a=\sum_{U}\sum_{i}a_{U_{i}}S_{U_{i}}=\sum_{U}a_{U},\quad
b=\sum_{V}\sum_{j}b_{V_{j}}T_{V_{j}}=\sum_{V}b_{V},\vspace{-.1cm}$

where $a_{U_{i}},b_{V_{j}}\in{\mathcal{M}}_{\mathbb{Q}+}$,
$S_{U_{i}},T_{V_{j}}\in\Pi^{Q}({\mathcal{S}})$ and
$a_{U_{i}}\perp^{Q}S_{U_{i}}$, $b_{V_{j}}\perp^{Q}T_{V_{j}}$, ${\rm
Supp}(S_{U_{i}})=U$, ${\rm Supp}(T_{V_{j}})=V$ (with the convention that ${\rm
Supp}(h)=0$ for $h\in{\mathcal{M}}_{\mathbb{Q}+}$).

Theorem 2.13 gives ${\rm Dep}(a_{U})\subset{\rm Dep}(a)$ and ${\rm
Dep}(b_{V})\subset{\rm Dep}(b)$. Since $a\perp^{Q}b$ means ${\rm
Dep}(a)\perp^{Q}{\rm Dep}(b)$, we have ${\rm Dep}(a_{U})\perp^{Q}{\rm
Dep}(b_{V})$, that is $a_{U}\perp^{Q}b_{V}$.

Now let $a_{U}\not=0$ and $b_{V}\not=0$.

  1. (i)

By Proposition 2.15, we have $U\subset{\rm Dep}(a)$ and $V\subset{\rm
Dep}(b)$. So $U\perp^{Q}V$ and $S_{U_{i}}\perp^{Q}T_{V_{j}}$;

  2. (ii)

From ${\rm Dep}(a_{U})\subset{\rm Dep}(a)$ and $V\subset{\rm Dep}(b)$, we
obtain $a_{U}\perp^{Q}T_{V_{j}}$. Similarly, $b_{V}\perp^{Q}S_{U_{i}}$;

  3. (iii)

By Proposition 2.16, there exist holomorphic germs $\tilde{a}_{U_{i}}$ and
$\tilde{b}_{V_{j}}$ with

${\rm Dep}(\tilde{a}_{U_{i}})\subset{\rm Dep}(a_{U}),\
\tilde{a}_{U_{i}}\perp^{Q}S_{U_{i}},\ {\rm Dep}(\tilde{b}_{V_{j}})\subset{\rm
Dep}(b_{U}),\ \tilde{b}_{V_{j}}\perp^{Q}T_{V_{j}},\vspace{-.2cm}$

such that

$a_{U}=\sum_{i}\tilde{a}_{U_{i}}S_{U_{i}},\quad
b_{V}=\sum_{j}\tilde{b}_{V_{j}}T_{V_{j}}.\vspace{-.2cm}$

From $a_{U}\perp^{Q}T_{V_{j}}$ and ${\rm Dep}(\tilde{a}_{U_{i}})\subset{\rm
Dep}(a_{U})$, we have $\tilde{a}_{U_{i}}\perp^{Q}T_{V_{j}}$. Likewise,
$\tilde{b}_{V_{j}}\perp^{Q}S_{U_{i}}$.

In summary, $\tilde{a}_{U_{i}},\tilde{b}_{V_{j}},S_{U_{i}},T_{V_{j}}$ are
mutually $Q$-orthogonal. Thus
$a_{U}b_{V}=\sum\limits_{i,j}\tilde{a}_{U_{i}}\tilde{b}_{V_{j}}S_{U_{i}}T_{V_{j}},$
is defined and gives an element in
${\mathcal{M}}_{\mathbb{Q}+}^{Q}(\Pi^{Q}(\widehat{{\mathcal{S}}})))$.
Therefore, $ab=\sum\limits_{U,V}a_{U}b_{V}$ is defined and lies in
${\mathcal{M}}_{\mathbb{Q}+}^{Q}(\Pi^{Q}(\widehat{{\mathcal{S}}}))$. ∎

In the subsequent examples, we implicitly fix an orthonormal basis
$\mathcal{E}$ with respect to an inner product $Q$ in $\mathbb{R}^{\infty}$,
only referring to these choices in the notation when necessary.

######  Example 3.8.

The set ${{\mathcal{F}}}^{\rm Ch}={\mathcal{F}}^{{\rm Ch},Q,\mathcal{E}}$ of
Chen fractions in Example 2.7 generates the unitary locality subalgebra
${\mathcal{M}}_{\mathbb{Q}}^{\rm
Ch}\coloneqq{\mathcal{M}}^{Q}_{\mathbb{Q}+}\left(\Pi^{Q}({\widehat{{\mathcal{F}}}}^{\rm
Ch})\right)$.

######  Example 3.9.

For a finite subset $J$ of $\mathbb{Z}_{>0}$, we set $z_{J}\coloneqq\sum_{i\in
J}z_{i}$. As considered by Speer [Sp2] (see § 5.2), a Feynman fraction is a
simplex fraction

(19) $\frac{1}{\prod_{J\in\mathcal{J}}z_{J}^{s_{J}}},s_{J}>0,\vspace{-.1cm}$

for a finite collection $\mathcal{J}$ of finite subsets of $\mathbb{N}$. With
similar notations of Example 3.8, the set of Feynman fractions and the
locality subalgebra it generates are denoted by

${{\mathcal{F}}}^{\rm Fe}\coloneqq{\mathcal{F}}^{{\rm
Fe},Q,\mathcal{E}},\quad{\mathcal{M}}_{\mathbb{Q}}^{\rm
Fe}\coloneqq{\mathcal{M}}_{\mathbb{Q}+}^{Q}\left(\Pi^{Q}({\widehat{{\mathcal{F}}}}^{\rm
Fe})\right).\vspace{-.2cm}$

###  3.2. Automorphism groups of simplex locality algebras

For a locality subalgebra ${\mathcal{A}}$ of
${({\mathcal{M}}_{\mathbb{Q}},\perp^{Q})}$, let ${\rm Aut}^{Q}({\mathcal{A}})$
denote the group of locality automorphisms of ${\mathcal{A}}$, following
Definition 3.5.

######  Proposition 3.10.

Let ${\mathcal{S}}$ be a set of simplex fractions and let
${\mathbb{Q}}\Pi^{Q}(\widehat{{\mathcal{S}}})$ be the unitary locality
subalgebra of $\mathbb{Q}\widehat{{\mathcal{F}}}^{Q}$ generated by
${\mathcal{S}}$. Let ${\rm Aut}_{\rm
Res}^{Q}({\mathbb{Q}}\Pi^{Q}(\widehat{{\mathcal{S}}}))$ be the set of unitary
locality algebra homomorphisms
$\varphi:{\mathbb{Q}}\Pi^{Q}(\widehat{{\mathcal{S}}})\to{\mathbb{Q}}\Pi^{Q}(\widehat{{\mathcal{S}}})$
with the property that, for any fraction $S$ in $\Pi^{Q}({\mathcal{S}})$,

(20) $\varphi\left(S\right)=S+\sum_{i}{a_{i}}\,S_{i},\vspace{-.2cm}$

where

$a_{i}\in\mathbb{Q},\,S_{i}\in\Pi^{Q}(\mathcal{S}),\,\text{\rm
p-ord}(S_{i})<\text{\rm p-ord}(S),\ {\rm Supp}(S_{i})\subsetneq{\rm
Supp}\left(S\right).\vspace{-.1cm}$

Then ${\rm Aut}_{\rm Res}^{Q}({\mathbb{Q}}\Pi^{Q}(\widehat{{\mathcal{S}}}))$
is a subgroup of ${\rm
Aut}^{Q}({\mathbb{Q}}\Pi^{Q}(\widehat{{\mathcal{S}}}))$.

######  Remark 3.11.

As a consequence of Theorem 3.15 yet to come, we have

(21) ${\rm Aut}_{\rm
Res}^{Q}({\mathbb{Q}}\Pi^{Q}(\widehat{{\mathcal{S}}}))=\Big{\\{}\left.\varphi\in{\rm
Aut}^{Q}({\mathbb{Q}}\Pi^{Q}(\widehat{{\mathcal{S}}}))\,\right|\,\varphi\,\text{preserves
the p-residue and d-residue}\Big{\\}}\vspace{-.1cm}$

which justifies the notation with subscript “Res”.

###### Proof.

Denote $R\coloneqq
R_{S}\coloneqq{\mathbb{Q}}\Pi^{Q}(\widehat{{\mathcal{S}}})$. We first prove
that $\varphi$ is one-to-one. Let

$f=\sum_{i}a_{i}S_{i}\in
R,a_{i}\in\mathbb{Q},S_{i}\in\Pi^{Q}({\mathcal{S}})\cup\\{1\\},\vspace{-.1cm}$

be nonzero. If $f$ is a constant in ${\mathbb{Q}}$, then $\varphi(f)=f\neq 0$.
If $f$ is not a constant, we group the terms of $f$ according to the gradation
in Eq. (9):

$f=c_{0}+\sum_{U\in\mathcal{U}}f_{U},\vspace{-.2cm}$

where $\mathcal{U}$ is a finite nonempty set of nonzero subspaces of
${\mathbb{R}}^{\infty}$ and $0\neq
f_{U}\in\mathbb{Q}\Pi^{Q}(\widehat{{\mathcal{S}}})$, for any $U$ in
$\mathcal{U}$, is a sum of fractions with supporting space $U$. Applying
$\varphi$ yields

$\varphi(f)=c_{0}+\sum_{U\in\mathcal{U}}\Big{(}f_{U}+\sum_{V\subsetneq
U}g_{V}\Big{)},\vspace{-.2cm}$

where $g_{V}$ is a sum (possibly zero) of fractions with supporting space $V$.
For a maximal element $U_{0}$ in $\mathcal{U}$, $f_{U_{0}}$ is the only
contribution in the above sum arising in $\varphi(f)$ to the component with
supporting space $U_{0}$ in the decomposition in Eq. (9). So $\varphi(f)=0$
implies $f_{U_{0}}=0$. This is a contradiction. It follows that
$\varphi(f)\neq 0$, which ends the proof of the injectivity.

To prove the surjectivity of $\varphi$, by the linearity of $\varphi$, we only
need to show that every element $f$ of $\Pi^{Q}(\widehat{{\mathcal{S}}})$ lies
in the range Im $\varphi$ of $\varphi$. Suppose this is not the case and let
$U_{0}\neq 0$ be minimal among the supporting spaces of elements in
$\Pi^{Q}(\widehat{{\mathcal{S}}})\backslash\text{Im}\varphi$. Let $f_{0}$ be
one of the simplex fractions in
$\Pi^{Q}(\widehat{{\mathcal{S}}})\backslash\text{Im}\varphi$ with supporting
space $U_{0}$. Then by Eq. (20) we have

$\varphi(f_{0})=f_{0}+\sum_{V\subsetneq U_{0}}f_{V},\vspace{-.1cm}$

where $f_{V}\in\mathbb{Q}(\Pi^{Q}(\widehat{{\mathcal{S}}}))$ is the sum of
simplex fractions with supporting space $V$. The space $U_{0}$ being minimal,
each $f_{V}$ lies in the image of $\varphi$. Therefore,
$f_{0}=\varphi(f_{0})-\sum\limits_{V\subsetneq U_{0}}f_{V}$ also lies in the
image of $\varphi$. This is a contradiction, showing that $\varphi$ is
surjective.

We finally prove that $\varphi^{-1}$ is a locality algebra homomorphism. We
first show that $\varphi^{-1}$ has the property in Eq. (20), that is, for any
$S\in\Pi^{Q}(\widehat{{\mathcal{S}}})$, we have

$\varphi^{-1}(S)=S+\sum_{i}h_{i}S_{i},\vspace{-.2cm}$

where each $S_{i}\in\Pi^{Q}(\widehat{{\mathcal{S}}})$ has smaller supporting
space and p-order than those of $S$. Assume that this were not the case, and
let $S$ have a minimal supporting space $U_{0}$ among the counterexamples.
Then

$\varphi(S)=S+\sum_{j}b_{j}T_{j},\vspace{-.2cm}$

where each simplex fraction $T_{j}\in\Pi^{Q}(\widehat{{\mathcal{S}}})$ has it
supporting space and p-order smaller than those of $S$. Applying
$\varphi^{-1}$ gives

(22) $S=\varphi^{-1}(S)+\sum_{j}b_{j}\varphi^{-1}(T_{j}).\vspace{-.2cm}$

The minimality of the supporting space of $S$ yields

$\varphi^{-1}(T_{j})=T_{j}+\sum_{jk}d_{jk}T_{jk},\vspace{-.2cm}$

where each $T_{jk}\in\Pi^{Q}(\widehat{{\mathcal{S}}})$ has its supporting
space and p-order smaller than those of $T_{j}$ and hence of $S$. Therefore,
Eq. (22) gives

$\varphi^{-1}(S)=S-\sum_{j}\varphi^{-1}(T_{j})=S-\sum_{j}b_{j}\left(T_{j}+\sum_{jk}d_{jk}T_{jk}\right),\vspace{-.2cm}$

which shows that $\varphi^{-1}(S)$ has the form in Eq. (20). This gives the
desired contradiction.

To check that $\varphi^{-1}$ is a locality map, we consider two linear
combinations $f,g$ in $R=\mathbb{Q}(\Pi^{Q}(\widehat{{\mathcal{S}}}))$ and
group the terms

$f=c+\sum_{U\in\mathcal{U}}f_{U}\quad{\rm and}\quad
g=d+\sum_{V\in\mathcal{V}}g_{V},\vspace{-.1cm}$

with $c,d$ in $\mathbb{Q}$ and nonzero sums
$f_{U},g_{V}\in\mathbb{Q}(\Pi^{Q}(\widehat{{\mathcal{S}}}))$ of fractions with
supporting spaces $U\in\mathcal{U}$ and $V\in{\mathcal{V}}$ respectively. We
proceed to show that $f\perp^{Q}g$ implies
$\varphi^{-1}(f)\perp^{Q}\varphi^{-1}(g)$.

Let $U$ be an element in $\mathcal{U}$ and $V$ an element in $\mathcal{V}$. By
Theorem 2.13, ${\rm Dep}\,f_{U}\subset{\rm Dep}\,f$ and ${\rm
Dep}\,g_{V}\subset{\rm Dep}\,g$ so that ${\rm Dep}\,f\perp^{Q}{\rm Dep}\,g$
implies ${\rm Dep}\,f_{U}\perp^{Q}{\rm Dep}\,g_{V}$ and hence
$f_{U}\perp^{Q}g_{V}.$ By Lemma 2.14, for the linear combination $f_{U}$
(resp. $g_{V}$) of simplex fractions with the same supporting space $U$ (resp.
$V$), we have $U={\rm Dep}(f_{U})$ (resp. $V={\rm Dep}(g_{V})$). Thus
$f_{U}\perp^{Q}g_{V}$ implies $U\perp^{Q}V$.

Since $\varphi^{-1}$ has the property in Eq. (20), we have ${\rm
Dep}(\varphi^{-1}(f_{U}))\subseteq{\rm Dep}(f_{U})$ and ${\rm
Dep}(\varphi^{-1}(g_{V})\subseteq{\rm Dep}(g_{V})$. Then
$\varphi^{-1}(f_{U})\perp^{Q}\varphi^{-1}(g_{V})$. Therefore, by the linearity
of $\varphi^{-1}$, we obtain $\varphi^{-1}(f)\perp^{Q}\varphi^{-1}(g)$, as
needed.

Finally for $f,g\in R$ with $f\perp^{Q}g$, we have
$\varphi^{-1}(f)\perp^{Q}\varphi^{-1}(g)$. Hence

$\varphi(\varphi^{-1}(f)\,\varphi^{-1}(g))=\varphi(\varphi^{-1}(f))\,\varphi(\varphi^{-1}(g))=fg.$

Therefore, applying $\varphi^{-1}$, we obtain
$\varphi^{-1}(f)\,\varphi^{-1}(g)=\varphi^{-1}(fg),$ showing that
$\varphi^{-1}$ is a locality homomorphism. ∎

###  3.3. Locality Galois groups

In this part we consider a locality subalgebra ${\mathcal{A}}$ of
$({\mathcal{M}}_{\mathbb{Q}},\perp^{Q})$ containing
${\mathcal{M}}_{\mathbb{Q}+}$.

######  Proposition 3.12.

For any locality morphism $\varphi:{\mathcal{A}}\to{\mathcal{A}}$ with
$\varphi|_{{\mathcal{M}}_{\mathbb{Q}+}}=\mathrm{Id}$, we have

${\rm Dep}(\varphi(f))\subset{\rm Dep}(f),\quad\forall f\in{\mathcal{A}}.$

###### Proof.

For any $\ell$ in ${\mathcal{L}}_{\mathbb{Q}}$ viewed as an element of
$(\mathbb{R}^{k})^{*}$ for some $k\geq 1$, if $\ell\perp^{Q}{\rm Dep}(f)$,
then $\ell\perp^{Q}f$, which implies that $\varphi(\ell)\perp^{Q}\varphi(f)$
since $\varphi$ is a locality map. Since $\ell$ is in
${\mathcal{M}}_{\mathbb{Q}+}$, we have $\varphi(\ell)=\ell$. Thus
$\ell\perp^{Q}{\rm Dep}(\varphi(x))$. So ${\rm Dep}(f)^{\top}\subseteq{\rm
Dep}(\varphi(x))^{\top}$ which yields the statement. ∎

In the sequel, we fix a set ${\mathcal{S}}\subseteq{\mathcal{F}}$ of simplex
fractions and let

(23)
${\mathcal{B}}\coloneqq{\mathcal{B}}({\mathcal{S}})\coloneqq\mathbb{Q}(\Pi^{Q}(\widehat{{\mathcal{S}}})),\quad{\mathcal{A}}\coloneqq{\mathcal{A}}({\mathcal{S}})\coloneqq{\mathcal{M}}_{\mathbb{Q}+}^{Q}\left(\Pi^{Q}(\widehat{{\mathcal{S}}})\right)\vspace{-.1cm}$

be the unitary locality subalgebra and
${\mathcal{M}}_{\mathbb{Q}+}^{Q}$-subalgebra of ${\mathcal{M}}_{\mathbb{Q}}$
generated by ${\mathcal{S}}$, defined in Propositions 3.6 and 3.7
respectively.

######  Definition 3.13.

Define a subset of $\mathrm{Aut}^{Q}({\mathcal{A}})$ by

${\rm
Gal}^{Q}({\mathcal{A}}/{\mathcal{M}}_{\mathbb{Q}+})\coloneqq\left\\{\varphi\in{\rm
Aut}^{Q}({\mathcal{A}})\,\left|\,\begin{array}[]{l}\varphi|_{{\mathcal{M}}_{\mathbb{Q}+}}=\mathrm{Id}\\\
\varphi\,\text{preserves the p-residue, d-residue}\\\ \text{and the locality
subalgebra}\,{\mathcal{B}}\end{array}\right.\right\\}\vspace{-.1cm}$

It will be called the locality Galois group of ${\mathcal{A}}$ over
${\mathcal{M}}_{\mathbb{Q}+}$, thanks to Theorem 3.15.

We next give a locality tensor product property of
${\mathcal{M}}_{\mathbb{Q}}$.

######  Proposition 3.14.

Let ${\mathcal{S}}\subseteq{\mathcal{F}}$ and, as in Eq. (23), define

${\mathcal{B}}\coloneqq{\mathcal{B}}({\mathcal{S}})\coloneqq\mathbb{Q}(\Pi^{Q}(\widehat{{\mathcal{S}}})),\quad{\mathcal{A}}\coloneqq{\mathcal{A}}({\mathcal{S}})\coloneqq{\mathcal{M}}_{\mathbb{Q}+}\left(\Pi^{Q}(\widehat{{\mathcal{S}}})\right).\vspace{-.1cm}$

  1. (i)

For each ${\mathbb{Q}}$-subspace $U$ of $\mathbb{R}^{\infty}$, let
${\mathcal{A}}_{U}\coloneqq{\mathcal{A}}\cap{\mathcal{M}}_{{\mathbb{Q}},U}$
and let ${\mathcal{B}}_{U}$ denote the linear span of simplex fractions in
${\mathcal{B}}$ with supporting space $U$. Let
${\mathcal{M}}_{{\mathbb{Q}}+}^{U}$ denote the space of holomorphic germs
whose dependent space is contained in

$U^{\perp^{Q}}\coloneqq\left\\{y\in{\mathcal{L}}(\mathbb{C}^{\infty})\,\left|\,y\perp^{Q}u,\forall
u\in U\right.\right\\}.$

Then we have the (inner) tensor product

${\mathcal{A}}_{U}={\mathcal{M}}_{{\mathbb{Q}}+}^{U}\otimes{\mathcal{B}}_{U},$

that is, ${\mathcal{M}}_{{\mathbb{Q}}+}^{U}$ and ${\mathcal{B}}_{U}$ are
linearly disjoint.

  2. (ii)

Let $(V,\top)$ be a locality vector space. Any pair of locality linear maps
$\varphi:{\mathcal{B}}\to V$ and $\psi:{\mathcal{M}}_{\mathbb{Q}+}\to V$
uniquely extends to a locality linear map

$\varphi\otimes^{Q}\psi:{\mathcal{A}}\to V.$

###### Proof.

(i) By Proposition 3.7,

(24) $f=\sum_{i}h_{i}S_{i}\,,\vspace{-.2cm}$

where $S_{i}\in\Pi^{Q}(\widehat{{\mathcal{S}}})$ with $\mathrm{Dep}(S_{i})=U$,
$h_{i}$ is holomorphic with dependent space contained in $U^{\perp^{Q}}$ and
hence $h_{i}\perp^{Q}S_{i}$. So
${\mathcal{A}}_{U}={\mathcal{M}}_{{\mathbb{Q}}+}^{U}{\mathcal{B}}_{U}$ as a
product of subsets.

To prove the disjointness, we more generally consider a linear combination

(25) $\sum_{i}h_{i}S_{i}=0,\vspace{-.3cm}$

where $\mathrm{Dep}(S_{i})=U$, $h_{i}$ is holomorphic with dependent space
contained in $U^{\perp^{Q}}$. Suppose that $\\{S_{i}\\}_{i}$ is linearly
independent, but $h_{i}\neq 0$ for all $i$ in the sum. Denote
$V=\sum_{i}\mathrm{Dep}(h_{i})$ which is a finite-dimensional subspace of
$U^{\perp^{Q}}$. Then $h_{i}$ is defined on $V$. Thus we can choose disjoint
sets of variables $\\{z_{k}\\}$ of $U$ and $\\{w_{\ell}\\}$ of $V$
respectively. From $h_{i}\neq 0$, there is $\\{w_{\ell}^{0}\\}$ such that
$h_{i}(\\{w_{\ell}^{0}\\})\neq 0$ for all $i$. Then Eq. (25) gives

$\sum_{i}h_{i}(\\{w_{\ell}^{0}\\})\,S_{i}=0,\vspace{-.2cm}$

showing that $\\{S_{i}\\}_{i}$ is linearly dependent. This gives the desired
contradiction.

(ii) By Proposition 3.7, ${\mathcal{A}}$ is linearly spanned by homogeneous
elements with respect to the grading by supporting space in the grading Eq.
(9). Thus ${\mathcal{A}}$ has the restricted grading

${\mathcal{A}}=\bigoplus_{U\subset\mathbb{R}^{\infty}}{\mathcal{A}}_{U}.$

Then we just need to show that $\varphi$ and $\psi$ uniquely define a locality
linear map

$(\varphi\otimes^{Q}\psi)_{U}:{\mathcal{A}}_{U}\to V\vspace{-.3cm}$

for each subspace $U$ of $\mathbb{R}^{\infty}$.

By Item (i), for any linear map $\varphi:{\mathcal{B}}_{U}\to V$ and
$\psi:{\mathcal{M}}_{{\mathbb{Q}}+}^{U}:\to V$, there is a unique linear map

(26) $(\varphi\otimes^{Q}\psi)_{U}:{\mathcal{A}}_{U}\to V,\quad
f\mapsto\sum_{i}\psi(h_{i})\varphi(S_{i})\vspace{-.3cm}$

for any element $f=\sum_{i}h_{i}S_{i}$ in ${\mathcal{A}}_{U}$, expressed in
the form in Eq. (24). Indeed, $(\varphi\otimes^{Q}\psi)_{U}$ is simply the
tensor product of the restriction of $\psi$ to
${\mathcal{M}}_{{\mathbb{Q}}+}^{U}$ and the restriction of $\varphi$ to
${\mathcal{B}}_{U}$. Taking the sum over all subspaces $U$ of
${\mathcal{L}}(\mathbb{R}^{\infty})$ including $U=0$, we have an extension
$\varphi\otimes^{Q}\psi$ of $\varphi$ and $\psi$ to ${\mathcal{A}}$. ∎

The remaining part of the section is devoted to the proof of the following
theorem which extends an element of ${\rm Aut}_{\rm Res}^{Q}({\mathcal{B}})$
to an element of
$\mathrm{Gal}^{Q}({\mathcal{A}}/{\mathcal{M}}_{\mathbb{Q}+})$.

######  Theorem 3.15.

Let ${\mathcal{S}}\subseteq{\mathcal{F}}$ and let ${\mathcal{A}}$ and
${\mathcal{B}}$ be as defined in Eq. (23).

  1. (i)

Any element $\varphi\in{\rm Aut}_{{\rm Res}}^{Q}({\mathcal{B}})$ (see
Proposition 3.10) uniquely extends to an element of
$\mathrm{Gal}^{Q}({\mathcal{A}}/{\mathcal{M}}_{\mathbb{Q}+})$ defined by

(27)
$\tilde{\varphi}\left(\sum_{i}h_{i}S_{i}\right)\coloneqq\sum_{i}h_{i}\varphi(S_{i})\vspace{-.2cm}$

for

(28)
$f=\sum_{i}h_{i}S_{i}\in{\mathcal{A}},h_{i}\in{\mathcal{M}}_{\mathbb{Q}+},S_{i}\in\Pi^{Q}(\widehat{{\mathcal{S}}})\vspace{-.2cm}$

as in Proposition 3.7.

  2. (ii)

The subset
$\mathrm{Gal}^{Q}({\mathcal{A}}/{\mathcal{M}}_{\mathbb{Q}+})\subseteq\mathrm{Aut}^{Q}({\mathcal{A}})$
is a subgroup. Restricting to ${\mathcal{B}}$ gives rise to a group
isomorphism

$\mathrm{Gal}^{Q}({\mathcal{A}}/{\mathcal{M}}_{\mathbb{Q}+})\cong{\rm
Aut}^{Q}_{{\rm Res}}({\mathcal{B}}).\vspace{-.2cm}$

###### Proof.

(28) Applying Proposition 3.14 with $\psi$ the identity map, for any linear
map $\varphi:{\mathcal{B}}_{U}\to{\mathcal{B}}_{U}$, there is a unique linear
map

(29)
$\tilde{\varphi}:{\mathcal{M}}_{{\mathbb{Q}},U}\to{\mathcal{M}}_{{\mathbb{Q}},U},\quad
f\mapsto\sum_{i}h_{i}\varphi(S_{i})\vspace{-.2cm}$

for any element $f=\sum_{i}h_{i}S_{i}$ in ${\mathcal{M}}_{{\mathbb{Q}},U}$,
expressed in the form in Eq. (24).

We next show that the $\tilde{\varphi}$ obtained this way has the form in Eq.
(27). Let $f=\sum_{i}h_{i}S_{i}$ as in Eq. (28). By grouping the terms
according to the supporting spaces of $S_{i}$ as in Eq. (9), we have

$f=\sum_{U\in\mathcal{U}}f_{U}\ \text{ with }\ f_{U}=\sum_{j}a_{Uj}S_{Uj},$

where $\mathcal{U}$ is a set of subspaces $U$ of $\mathbb{R}^{\infty}$ for
which $f_{U}\neq 0$, and for each $U\in\mathcal{U}$, we have

$S_{Uj}\in\Pi^{Q}({\mathcal{S}}),\quad{\rm Dep}(S_{Uj})={\rm
Supp}(S_{Uj})=U,\quad{\rm Dep}(a_{Uj})\perp^{Q}{\rm Dep}(S_{Uj})$

and each term $a_{Uj}S_{Uj}$ is one of the terms in $f=\sum_{i}h_{i}S_{i}$.
Thus $f_{U}$ is in ${\mathcal{M}}_{{\mathbb{Q}},U}$ and we can apply Eq. (29)
and obtain

$\tilde{\varphi}(f_{U})=\sum_{i}a_{Uj}\varphi(S_{Uj}),\vspace{-.1cm}$

which takes the form in Eq. (27). Hence so is $\tilde{\varphi}(f)$. This is
what we want.

The fact that $\tilde{\varphi}$ preserves the p-residue in Eq. (10) and the
d-residue in Eq. (11) follows from the definition and the special form of
$\varphi$ on ${\mathcal{B}}$.

For
$f=\sum_{i}h_{i}S_{i}\in{\mathcal{A}}={\mathcal{M}}_{\mathbb{Q}+}(\Pi^{Q}({\mathcal{S}}))$
with $h_{i}\in{\mathcal{M}}_{\mathbb{Q}+},S_{i}\in{\mathcal{S}}$, the
p-residue $\text{\rm p-res}(f)$ of $f$ is of the form
$\sum_{i}^{\prime}h_{i}(0)S_{i}$ where the sum is over simplex fractions
$S_{i}$ in $f$ with the highest order. By the definition of $\tilde{\varphi}$,
the sum $\sum_{i}^{\prime}h_{i}S_{i}$ is still the part of
$\tilde{\varphi}(f)$ with the highest order. Therefore, $\text{\rm
p-res}(\tilde{\varphi}(f))=\text{\rm p-res}(f)$.

The same argument, applied to the dimensions of supporting spaces of the polar
germs, shows that $\tilde{\varphi}$ preserves the d-residues.

We next check that $\widetilde{\varphi}$ is a locality
${\mathcal{M}}_{\mathbb{Q}+}$-algebra homomorphism. For $a,b\in{\mathcal{A}}$
with $a\perp^{Q}b$, as in the proof of Proposition 3.6, we can write them as

$a=\sum_{U}\sum_{i}h_{U_{i}}S_{U_{i}},\quad
b=\sum_{V}\sum_{j}g_{V_{j}}T_{V_{j}}\vspace{-.2cm}$

such that

$h_{U_{i}}\not=0,g_{V_{j}}\not=0,\
\\{h_{U_{i}},S_{U_{i}}\\}\perp\\{g_{V_{j}},T_{V_{j}}\\}.\vspace{-.1cm}$

By the special form of $\varphi$, we have

${\rm Dep}(\varphi(S_{U_{i}}))=U,\quad{\rm Dep}(\varphi(T_{Vi}))=V.$

Then it follows from the definition of $\tilde{\varphi}$ that
$\tilde{\varphi}(a)\perp^{Q}\tilde{\varphi}(b).$ Moreover (treating $h_{0}$ as
$h_{U}S_{U}$ for $U=0$ and the same for $g_{0}$),

$\tilde{\varphi}(ab)=\tilde{\varphi}\left(\sum_{U,V}h_{U_{i}}g_{V_{j}}S_{U_{i}}T_{V_{j}}\right)=\sum_{U,V}h_{U_{i}}g_{V_{j}}\varphi(S_{U_{i}}T_{V_{j}})=\sum_{U,V}h_{U_{i}}g_{V_{j}}\varphi(S_{U_{i}})\varphi(T_{V_{j}})=\tilde{\varphi}(a)\tilde{\varphi}(b).\vspace{-.1cm}$

By construction, the extension $\varphi\mapsto\tilde{\varphi}$ is functorial:

$\widetilde{\varphi\psi}=\tilde{\varphi}\tilde{\psi},\quad\widetilde{\rm
id}_{\mathcal{B}}={\rm id}_{\mathcal{A}}.$

So for any $\varphi\in{\rm Aut}_{{\rm Res}}^{Q}({\mathcal{B}})$,
$\tilde{\varphi}$ is a linear bijection. The functorial property also shows
that $\tilde{\varphi}\,\widetilde{\varphi^{-1}}={\rm id}_{\mathcal{A}}$. So
$\tilde{\varphi}^{-1}=\widetilde{\varphi^{-1}}$. Thus
$\widetilde{\varphi}^{-1}$ is also a locality
${\mathcal{M}}_{\mathbb{Q}+}$-algebra homomorphism. Thus $\tilde{\varphi}$ is
in $\mathrm{Gal}^{Q}({\mathcal{A}}/{\mathcal{M}}_{\mathbb{Q}+})$ for all
$\varphi\in{\rm Aut}_{{\rm Res}}^{Q}({\mathcal{B}})$. By
$\widetilde{\varphi\,\psi}=\tilde{\varphi}\tilde{\psi},\quad\tilde{\varphi}^{-1}=\widetilde{\varphi^{-1}},$
the image of the map

(30) $\Psi:{\rm Aut}^{Q}_{{\rm
Res}}({\mathcal{B}})\to\mathrm{Gal}^{Q}({\mathcal{A}}/{\mathcal{M}}_{\mathbb{Q}+}):\varphi\mapsto\tilde{\varphi},\vspace{-.1cm}$

is a subgroup of $\mathrm{Aut}^{Q}({\mathcal{A}})$.

(ii) The map $\Psi$ defined in Eq. (30) is clearly injective and its image is
in $\mathrm{Gal}^{Q}({\mathcal{A}}/{\mathcal{M}}_{\mathbb{Q}+})$. Now for any
$g\in\mathrm{Gal}^{Q}({\mathcal{A}}/{\mathcal{M}}_{\mathbb{Q}+})$, since it
preserves ${\mathcal{B}}$, for any $S\in\Pi^{Q}({\mathcal{S}})$,

$g(S)=\sum_{i}a_{i}S_{i},\vspace{-.2cm}$

with $a_{i}\in\mathbb{Q}$, $S_{i}\in\Pi^{Q}({\mathcal{S}})$.

Note that the p-residue of $g(S)$ equals to the p-residue of $S$ which is just
$S$. So we can write

$g(S)=\sum a_{i}S_{i}=\sum_{{\text{\rm p-ord}(S_{i})}={\text{\rm
p-ord}}(S)}a_{i}S_{i}+\sum_{{\text{\rm p-ord}(S_{i})}<{\text{\rm
p-ord}}(S)}a_{i}S_{i}$

and

$\sum_{{\text{\rm p-ord}(S_{i})}={\text{\rm p-ord}}(S)}a_{i}S_{i}=S.$

So

$g(S)=S+\sum_{{\text{\rm p-ord}(S_{i})}<{\text{\rm p-ord}}(S)}a_{i}S_{i}.$

Also, the d-residue of $S$ is also $S$. Write

$g(S)=S+\sum_{U}S_{U}=S+\sum_{U}\sum_{i}a_{U_{i}}S_{U_{i}},$

with $\dim U<\dim{\rm Supp}(S)$, $0\neq
S_{U}=\sum_{i}a_{U_{i}}S_{U_{i}}\in\mathbb{Q}(\Pi^{Q}({\mathcal{S}}))$ and
${\rm Supp}(S_{U_{i}})=U.$ By Lemma 2.14, Proposition 3.12 and Theorem 2.13,
we obtain

$U={\rm Dep}(S_{U})\subseteq{\rm Dep}(g(S))={\rm Dep}(S)={\rm Supp}(S).$

Hence ${\rm Supp}(S_{U_{i}})\subsetneq{\rm Supp}(S)$. Thus $g|_{\mathcal{B}}$
is in ${\rm Aut}^{Q}_{{\rm Res}}({\mathcal{B}})$ and we have
$\Psi(g|_{\mathcal{B}})=g,$ giving us the desired isomorphism. ∎

##  4\. Locality polynomial algebras generated by Chen and Speer fractions

In this section, we show that the locality subalgebras of ${\mathcal{F}}$
generated by Chen fractions of Example 2.7 and by the more general class of
Speer fractions over $\mathbb{Q}$ are both locality polynomial algebras,
defined to be the following locality version of polynomial algebras.

######  Definition 4.1.

Let $(A,\top)$ be a locality $K$-algebra. Let $X$ be a subset of $A$.

  1. (i)

A locality monomial built from $X$ is a product $x_{1}\cdots x_{r}$ where
$x_{i}\top x_{j}$ for $i\neq j$.

  2. (ii)

The set $X$ is called locality algebraically independent if distinct locality
monomials built from $X$ are linearly independent.

  3. (iii)

The set $X$ is called a locality generating set of $(A,\top)$ if the only
locality subalgebra of $(A,\top)$ containing $X$ is $A$ itself.

  4. (iv)

The locality algebra $(A,\top)$ is called a locality polynomial algebra
generated by $X$ if $X$ is locality algebraically independent, and is a
locality generating set of $(A,\top)$. Then we denote $A\cong{K}_{\top}[X].$

###  4.1. Locality shuffle algebras as locality polynomial algebras

Let us first recall how shuffle algebras can be viewed as polynomially
generated algebras. For a commutative ring $K$ and a set $X$, let $A={K}X$ be
the linear space with a basis $X$, $W(X)$ be the set of words with letters in
$X$ including the empty word. The shuffle algebra on $X$ is the space

$\mathrm{Sh}({K}X):={K}W(X),$

equipped with the shuffle product X . More precisely,

$(\alpha_{1}\vec{\alpha}^{\prime})\,\mbox{\bf\scyr
X}\,(\beta_{1}\vec{\beta}^{\prime})\coloneqq\alpha_{1}(\vec{\alpha}^{\prime}\,\mbox{\bf\scyr
X}\,(\beta_{1}\vec{\beta}^{\prime}))+\beta_{1}((\alpha_{1}\vec{\alpha}^{\prime})\,\mbox{\bf\scyr
X}\,\vec{\beta}^{\prime}),\quad\alpha_{1},\beta_{1}\in
X,\vec{\alpha}^{\prime},\vec{\beta}^{\prime}\in W(X),$

with the initial condition $1\,\mbox{\bf\scyr
X}\,\vec{\alpha}=\vec{\alpha}=\vec{\alpha}\,\mbox{\bf\scyr X}\,1.$

For a well-ordered set $(X,\leq)$, equip $W(X)$ with the lexicographic order
$\leq_{\mathrm{lex}}$. A word $w$ in $W(X)$ is called a Lyndon word if $w$ is
the smallest among all its rotations. Equivalently, $w$ is Lyndon if $w$ is
lexicographically smaller than all of its suffixes: $w=uv$ with $u,v\neq 1$
implies $w\leq_{\mathrm{lex}}v$.

######  Theorem 4.2.

[CFL, Ra] Let $(X,\leq)$ be a well-ordered set.

  1. (i)

$($Chen-Fox-Lyndon$)$ Any word $w$ in $W(X)$ has a unique factorisation

(31) $w=w_{1}^{i_{1}}\cdots w_{k}^{i_{k}}$

where $w_{1}>_{\rm lex}\cdots>_{\rm lex}w_{k}$ are Lyndon words and
$i_{1},\ldots,i_{k}\geq 1$.

  2. (ii)

$($Radford$)$ Let ${K}$ be a ${\mathbb{Q}}$-algebra. The set $Lyn(X)$ of
Lyndon words on $X$ is an algebraically independent generating set of
$\mathrm{Sh}({K}X)$. Thus $\mathrm{Sh}({K}X)\cong{K}[\mathrm{Lyn}(X)]$ is a
polynomial algebra. In fact, with the factorisation $w=w_{1}^{i_{1}}\cdots
w_{k}^{i_{k}}$ in Eq. (31), we have

(32) $w=\frac{1}{i_{1}!\cdots i_{k}!}w_{1}^{\,\mbox{\bf\scyr
X}\,i_{1}}\,\mbox{\bf\scyr X}\,\cdots\,\mbox{\bf\scyr
X}\,w_{k}^{\,\mbox{\bf\scyr X}\,i_{k}}+\text{smaller terms}.$

From a set $U$ we build a set

$\overline{{U}}\coloneqq\\{x_{0}\\}\sqcup\\{x_{u}\ |\ u\in{U}\\}.$

For a well-ordered set $(U,\leq)$, we then define a well-order $\leq$ on
$\overline{U}$ by imposing

(33) $x_{0}<x_{u},\ \text{ and }x_{u}\leq x_{v}\Leftrightarrow u\leq v,\
\forall u,v\in U.$

Denote

$W_{1}(\overline{{U}})\coloneqq\\{1\\}\cup\prod_{u\in
U}W(\overline{{U}})x_{u}\text{ and
}\mathrm{Sh}_{1}({K}\overline{{U}})\coloneqq{K}W_{1}(\overline{{U}})={K}\bigoplus\left(\bigoplus_{u\in{U}}\mathrm{Sh}({K}\overline{{U}})x_{u}\right).$

Note that $\mathrm{Sh}_{1}({K}\overline{U})$ is closed under the shuffle
product, and the factorisation of $w$ in $W_{1}(\overline{{U}})$ given in Eq.
(31) has its factors in $W_{1}(\overline{U})$. Thus we obtain (see also [Zh, §
3.3.1])

######  Proposition 4.3.

Let
$\mathrm{Lyn}_{1}(\overline{U})\coloneqq\mathrm{Lyn}(\overline{U})\backslash\\{x_{0}\\}$,
that is, the set of Lyndon words in $\overline{U}$ that do not end with
$x_{0}$. Then $\mathrm{Sh}_{1}({K}\overline{U})$ is a subalgebra of
$\mathrm{Sh}({K}\overline{U})$ and is a polynomial algebra generated by
$\mathrm{Lyn}_{1}(\overline{U})$.

We now extend these constructions to the locality setting.

For a locality set $(X,\top)$, let $W_{\top}(X)$ denote the subset of $W(X)$
consisting of locality words, namely the words $w=w_{1}\cdots w_{k}$ in which
$w_{i}\top w_{j},1\leq i\neq j\leq k$, plus the empty word. Let

$\mathrm{Sh}_{\top}({K}X)\coloneqq{K}W_{\top}(X).$

For $w=w_{1}\cdots w_{k}$ and $v=v_{1}\cdots v_{\ell}$ in $W_{\top}(X)$,
define

(34) $w\top v\Longleftrightarrow w_{i}\top v_{j},\forall 1\leq i\leq k,1\leq
j\leq\ell.$

Thus for $w,v$ in $W_{\top}(X)$ with $w\top v$, the word $wv$ also lies in
$W_{\top}(X)$. Since a shuffle of $w$ and $v$ is obtained from $wv$ by
permuting the factors and hence still lies in $W_{\top}(X)$, the shuffle
product $w\,\mbox{\bf\scyr X}\,v$ lies in $\mathrm{Sh}_{\top}({K}X)$. It
follows that $\mathrm{Sh}_{\top}({K}X)$ is a locality algebra.

For a well-ordered set $(X,\leq)$ equipped with a locality relation $\top$,
the Lyndon words in $W_{\top}(X)$ are called locality Lyndon words. The
following statement enhances Theorem 4.2 to a locality setting.

######  Theorem 4.4.

Let $(X,\leq)$ be a well-ordered set equipped with a locality relation $\top$.

  1. (i)

(Locality Chen-Fox-Lyndon Theorem) Any word $w$ in $W_{\top}(X)$ has a unique
factorisation

(35) $w=w_{1}^{i_{1}}\cdots w_{k}^{i_{k}}$

where $w_{1}>_{\rm lex}\cdots>_{\rm lex}w_{k}$ are locality Lyndon words and
$i_{1},\ldots,i_{k}\geq 1$.

  2. (ii)

(Locality Radford Theorem) Let ${K}$ be a ${\mathbb{Q}}$-algebra. The set
$Lyn_{\top}(X)$ of locality Lyndon words on $X$ is a locality algebraically
independent generating set $($in the sense of Definition 4.1$)$ of the
locality algebra $\mathrm{Sh}_{\top}({K}X)$. Thus
$\mathrm{Sh}_{\top}({K}X)\cong{K}_{\top}[\mathrm{Lyn}_{\top}(X)]$ is a
locality polynomial algebra.

###### Proof.

(35) In the factorisation $w=w_{1}^{i_{1}}\cdots w_{k}^{i_{k}}$ of a locality
word $w$ into Lyndon words in Theorem 4.2(31), each $w_{i}$ is still a
locality word, giving us the existence of the factorisation in Eq. (35). The
uniqueness of the factorisation follows from the uniqueness of the
factorisation in Eq. (31).

(ii) If $w$ is local, then by Item (35) and Theorem 4.2(ii),

$w=w_{1}^{i_{1}}\,\mbox{\bf\scyr X}\,w_{2}^{i_{2}}\,\mbox{\bf\scyr
X}\,\cdots\,\mbox{\bf\scyr X}\,w_{k}^{i_{k}}+\text{smaller terms},$

that is, $w$ can be generated by locality Lyndon words modulo smaller terms.
The smaller terms are obtained from $w$ by permuting the letters in $w$, and
hence are again local. Thus as in the nonlocality case, an induction can be
applied to show that $\mathrm{Sh}_{\top}({K}X)$ is spanned by
$\mathrm{Lyn}_{\top}(X)$. Locality algebraic independence of the set
$\mathrm{Lyn}_{\top}(X)$ is automatic since it is a subset of the
algebraically independent set $\mathrm{Lyn}(X)$ and locality algebraic
independence is weaker than algebraic independence. Therefore, locality Lyndon
words are locality polynomial generators of $\mathrm{Sh}_{\top}({K}X)$. ∎

Now let $(U,\leq)$ be a well-ordered set equipped with an irreflexive locality
relation $\perp$, i.e., $u\not\perp u$ for any $u$ in $U$.

Then $(\overline{{U}},\leq)$ is a well-ordered set by Eq. (33), and is
equipped with a locality relation $\top$ defined by

(36) $x_{0}\top x_{u},\forall u\in U\cup\\{x_{0}\\}\text{ and }x_{u}\top
x_{v}\text{ whenever }u\perp v,\quad\forall u,v\in U.$

Denote

$W_{1,\top}(\overline{U})\coloneqq W_{1}(\overline{U})\cap
W_{\top}(\overline{U}),\quad\mathrm{Sh}_{1,\top}({K}\overline{U})\coloneqq{K}W_{1,\top}(\overline{U}).$

######  Corollary 4.5.

Let $(U,\leq)$ be a well-ordered set equipped with an irreflexive locality
relation $\perp$.

  1. (i)

Any word $w$ in $W_{\top}(\overline{U})$ admits a unique factorisation

(37) $w=w_{1}\cdots w_{k}x_{0}^{r},$

where $w_{1}>_{\rm lex}\cdots>_{\rm lex}w_{k}>_{\rm lex}x_{0}$ are locality
Lyndon words in $W_{\top}(\overline{U})$, and $k,r\geq 0$.

  2. (ii)

Given a ${\mathbb{Q}}$-algebra ${K}$, the locality algebra
$\mathrm{Sh}_{\top}({K}\overline{U})$ is a locality polynomial algebra
generated by the set $\mathrm{Lyn}_{\top}(\overline{U})$ of locality Lyndon
words on $\overline{U}:$
$\mathrm{Sh}_{\top}({K}\overline{U})\cong{K}_{\top}[\mathrm{Lyn}_{\top}(\overline{U})].$

  3. (iii)

The subspace $\mathrm{Sh}_{1,\top}({K}\overline{U})$ of
$\mathrm{Sh}_{\top}({K}\overline{U})$ is a locality subalgebra. It is also a
locality polynomial algebra generated by
$\mathrm{Lyn}_{1,\top}(\overline{U})\coloneqq\mathrm{Lyn}_{1}(\overline{U})\cap\mathrm{Lyn}_{\top}(\overline{U})$.

###### Proof.

(37) Since $x_{0}$ is the smallest locality Lyndon word in
$W_{\top}(\overline{U})$, it must appear at the end of factorisation in Eq.
(35), giving us $w=w_{1}^{i_{1}}\cdots w_{k}^{i_{k}}x_{0}^{r}$ with
$w_{1}>_{\rm lex}w_{2}>_{\rm lex}\cdots>_{\rm lex}w_{k}>_{\rm lex}x_{0}$.
Further, a locality Lyndon word $w_{i}>_{\rm lex}x_{0}$ must have a factor
$x_{u}$ for some $u\in U$. So the irreflexivitiy of the locality and the
locality of $w$ implies that $i_{1}=\cdots=i_{k}=1$ and $w_{i}\top w_{j}$ for
$1\leq i\neq j\leq k$. The uniqueness of this factorisation follows from the
uniqueness of the factorisation in Theorem 4.4.(ii).

(ii) This follows from Theorem 4.4.(35).

(iii) The proof goes as for Proposition 4.3. ∎

###  4.2. Applications to ordered fractions

For the rest of this section, we fix an orthonormal basis ${\mathcal{E}}$ of
${\mathbb{R}}^{\infty}$ with respect to the inner product $Q$. For the sake of
simplicity, this choice, which we make once for all, is suppressed in most of
the notations.

Let $U$ be a countable set and let

(38) $L:U\to{\mathcal{L}}_{\mathbb{Q}},\quad u\mapsto L_{u},u\in U,$

be a map with values in the space ${\mathcal{L}}_{\mathbb{Q}}$ of linear forms
defined as in Eq. (4) with $K={\mathbb{Q}}$, which defines a family of linear
forms parameterised by $U$. For $u_{i}$ in $U$, $s_{i}\geq 1,1\leq i\leq k$,
define the ordered fraction (with respect to $L$)

${\mathfrak{f}}^{L}\Big{(}\begin{array}[]{c}\scriptstyle{s_{1},\ldots,s_{k}}\\\
\scriptstyle{{u_{1}},\ldots,{u_{k}}}\end{array}\Big{)}\coloneqq\frac{1}{L_{u_{1}}^{s_{1}}(L_{u_{1}}+L_{u_{2}})^{s_{2}}\cdots(L_{u_{1}}+\cdots+L_{u_{k}})^{s_{k}}}.$

Define the set of ordered fractions (with respect to $L$)

(39)
${\mathcal{F}}^{L}\coloneqq\bigg{\\{}{\mathfrak{f}}^{L}\Big{(}\begin{array}[]{c}\scriptstyle{s_{1},\ldots,s_{k}}\\\
\scriptstyle{{u_{1}},\ldots,{u_{k}}}\end{array}\Big{)}\,\bigg{|}s_{i}\geq
1,u_{i}\in U,1\leq i\leq k,k\geq 0\bigg{\\}}\subseteq{\mathcal{F}}.$

Note that ${\mathbb{Q}}{\mathcal{F}}^{L}$ is the ${\mathbb{Q}}$-subspace
spanned by ${\rm im}(L)$.

######  Example 4.6.

When $L=L_{{\rm wCh}}:\mathbb{Z}_{>0}\to{\mathcal{L}}_{\mathbb{Q}}$ is given
by $L(u)=z_{u}$, then

(40)
${\mathfrak{f}}^{L}\Big{(}\begin{array}[]{c}\scriptstyle{s_{1},\ldots,s_{k}}\\\
\scriptstyle{{u_{1}},\ldots,{u_{k}}}\end{array}\Big{)}={\mathfrak{f}}^{{\rm
wCh}}\Big{(}\begin{array}[]{c}\scriptstyle{s_{1},\ldots,s_{k}}\\\
\scriptstyle{{u_{1}},\ldots,{u_{k}}}\end{array}\Big{)}$

from Example 3.8 and ${\mathcal{F}}^{L}={\mathcal{F}}^{{\rm wCh}}$ is the set
of weak Chen fractions (called MZV fractions in [GX] for applications to
multiple zeta values).

Note that in contrast with a weak Chen fraction, a Chen fraction requires
$u_{i}\neq u_{j}$ for $i\neq j$. On the other hand, the fraction
$\frac{1}{(z_{1}+z_{2})(z_{2}+z_{3})}$ is not an ordered fraction with respect
to this $L$.

######  Proposition 4.7.

Let $U$ be countable and let $L:U\to{\mathcal{L}}_{\mathbb{Q}}$ be a map. Then
the map

(41)
$\Phi=\Phi^{L}:(\mathrm{Sh}_{1}({\mathbb{Q}}\overline{U}),\,\mbox{\bf\scyr
X}\,)\to{\mathbb{Q}}\widehat{{\mathcal{F}}}^{L},\quad
x_{0}^{s_{1}-1}x_{u_{1}}\cdots
x_{0}^{s_{k}-1}x_{u_{k}}\mapsto{\mathfrak{f}}^{L}\Big{(}\begin{array}[]{c}\scriptstyle{s_{1},\ldots,s_{k}}\\\
\scriptstyle{u_{1},\ldots,u_{k}}\end{array}\Big{)},$

is an algebra homomorphism. Here the multiplication on
${\mathbb{Q}}\widehat{{\mathcal{F}}}^{L}$ is the natural one in
${\mathcal{M}}_{\mathbb{Q}}$.

###### Proof.

When the map $L$ is given by

$L_{\rm wCh}:\mathbb{Z}_{>0}\to{\mathcal{L}}_{\mathbb{Q}},\quad u\mapsto
z_{u},$

then ${\mathcal{F}}^{L}={\mathcal{F}}^{{\rm wCh}}$ with ${\mathcal{F}}^{{\rm
wCh}}$ defined in Eq. (40) is the set of MZV fractions, in which case the
conclusion, for $\Phi=\Phi^{{\rm
wCh}}:\mathrm{Sh}_{1}(\mathbb{Q}\overline{\mathbb{Z}_{>0}})\to{\mathbb{Q}}\widehat{{\mathcal{F}}}^{{\rm
wCh}}$ follows from [GX, Eqs. (7),(8) and Theorem 2.1]. For a general $U$ and
$L$, on the grounds of the countability of $U$, we can fix a bijection
$\theta:U\to\mathbb{Z}_{>0}$ and thus an algebra isomorphism

$\theta:\mathrm{Sh}_{1}(\mathbb{Q}\overline{U})\to\mathrm{Sh}_{1}(\mathbb{Q}\overline{\mathbb{Z}_{>0}}).$

Also note that the change of variables $z_{i}\mapsto
L_{i},i\in{\mathbb{Z}}_{>0},$ gives rise to an algebra homomorphism

$\displaystyle\eta:{\mathbb{Q}}\widehat{{\mathcal{F}}}^{{\rm
wCh}}\to{\mathbb{Q}}\widehat{{\mathcal{F}}}^{L},$
$\displaystyle{\mathfrak{f}}^{{\rm
wCh}}\Big{(}\begin{array}[]{c}\scriptstyle{s_{1},\ldots,s_{k}}\\\
\scriptstyle{u_{1},\ldots,u_{k}}\end{array}\Big{)}\longmapsto{\mathfrak{f}}^{L}\Big{(}\begin{array}[]{c}\scriptstyle{s_{1},\ldots,s_{k}}\\\
\scriptstyle{u_{1},\ldots,u_{k}}\end{array}\Big{)}=\frac{1}{L_{u_{1}}^{s_{1}}(L_{u_{1}}+L_{u_{2}})^{s_{2}}\cdots(L_{u_{1}}+\cdots+L_{u_{k}})^{s_{k}}}.$

Then $\Phi^{L}$ is just the composition $\eta\circ\Phi^{{\rm
wCh}}\circ\theta$. ∎

######  Remark 4.8.

The algebra homomorphism $\Phi$ is not injective. For example,
${\mathfrak{f}}\Big{(}\begin{array}[]{c}\scriptstyle{2}\\\
\scriptstyle{u_{1}}\end{array}\Big{)}=\frac{1}{L_{u_{1}}^{2}}$ and
${\mathfrak{f}}\Big{(}\begin{array}[]{c}\scriptstyle{1,1}\\\
\scriptstyle{u_{1},u_{1}}\end{array}\Big{)}=\frac{1}{2}\frac{1}{L_{u_{1}}^{2}}$.
Consequently, although the shuffle algebra
$\mathrm{Sh}_{1}({\mathbb{Q}}\overline{U})$ is a polynomial algebra on the
Lyndon words, the same cannot be said of
${\mathbb{Q}}\widehat{{\mathcal{F}}}^{L}$. As we will see below, this defect
can be remedied under a locality condition.

Let $(U,\top)$ be a countable locality set and let
$L:U\to{\mathcal{L}}_{\mathbb{Q}}$ be as above. Consider the subset

(43)
${\mathcal{F}}^{L}_{\top}\coloneqq\left\\{\left.{\mathfrak{f}}^{L}\Big{(}\begin{array}[]{c}\scriptstyle{s_{1},\ldots,s_{k}}\\\
\scriptstyle{u_{1},\ldots,u_{k}}\end{array}\Big{)}\ \right|\ s_{i}\geq
1,u_{i}\in{U},u_{i}\top u_{j}\in U,1\leq i\not=j\leq k,k\geq
0\right\\}\subseteq{\mathcal{F}}^{L}$

and the ${\mathbb{Q}}$-subspace ${\mathbb{Q}}{\mathcal{F}}^{L}_{\top}$ of
${\mathbb{Q}}{\mathcal{F}}^{L}$.

######  Theorem 4.9.

Let $(U,\leq,\top)$ be a countable well-ordered set with an irreflexive
locality relation $\perp$. Suppose that the map
$L:(U,\top)\to({\mathcal{L}}_{\mathbb{Q}},\perp^{Q})$ defined in Eq. (38) is a
locality map in the sense of Eq. (15)$:$ for $x,y\in U$, if $x\top y$, then
$L_{x}\perp^{Q}L_{y}.$ Then

  1. (i)

the set ${\mathcal{F}}^{L}_{\top}$ is linearly independent;

  2. (ii)

the algebra homomorphism $\Phi$ in Eq. (41) restricts to an isomorphism of
locality algebras

(44)
$\Phi_{\top}:\mathrm{Sh}_{1,\top}(\overline{U})\cong\mathbb{Q}{\mathcal{F}}^{L}_{\top},\quad
w=x_{0}^{s_{1}-1}x_{u_{1}}\cdots
x_{0}^{s_{k}-1}x_{u_{k}}\mapsto{\mathfrak{f}}^{L}\Big{(}\begin{array}[]{c}\scriptstyle{s_{1},\ldots,s_{k}}\\\
\scriptstyle{u_{1},\ldots,u_{k}}\end{array}\Big{)}.\vspace{-.2cm}$

  3. (iii)

The locality algebra ${\mathbb{Q}}{\mathcal{F}}^{L}_{\top}$ is a locality
polynomial algebra.

###### Proof.

(i) The first assertion follows from the facts that the supporting cones of
all ordered fractions are projectively properly positioned and that by [GPZ3,
Proposition 3.6] recalled in Proposition 2.3, a projectively properly
positioned family of simplicial fractions is linearly independent.

(ii) By the assumption on $(U,\leq,\top)$, a word
$w=x_{0}^{s_{1}-1}x_{u_{1}}\cdots x_{0}^{s_{k}-1}x_{u_{k}}$ in
$\mathrm{Sh}({\mathbb{Q}}\overline{U})$ is local if and only if $u_{i}\top
u_{j},i\not=j$. So $w$ is local if and only if
$\Phi_{\top}(w)={\mathfrak{f}}^{L}\Big{(}\begin{array}[]{c}\scriptstyle{s_{1},\ldots,s_{k}}\\\
\scriptstyle{u_{1},\ldots,u_{k}}\end{array}\Big{)}$ lies in
${\mathcal{F}}^{L}_{\top}$. Thus by Item (i), $\Phi_{\top}$ sends a linear
basis of $\mathrm{Sh}_{1,\top}({\mathbb{Q}}\overline{U})$ to a linear basis of
${\mathbb{Q}}{\mathcal{F}}_{\top}^{L}$ and is therefore a linear isomorphism.

The linear map also preserves the locality. Indeed, for

$w_{1}=x_{0}^{s_{1}-1}x_{u_{1}}\cdots x_{0}^{s_{k}-1}x_{u_{k}},\
w_{2}=x_{0}^{t_{1}-1}x_{v_{1}}\cdots x_{0}^{t_{\ell}-1}x_{v_{\ell}}\in
W_{1}(\overline{U}),\vspace{-.2cm}$

we have

$w_{1}\top
w_{2}\Leftrightarrow\\{u_{1},\ldots,u_{k}\\}\top\\{v_{1},\ldots,v_{\ell}\\}\Leftrightarrow\\{L_{u_{1}},\ldots,L_{u_{k}}\\}\top\\{L_{v_{1}},\ldots,L_{v_{\ell}}\\}$
$\Leftrightarrow{\rm Supp}(\Phi(w_{1}))\perp^{Q}{\rm
Supp}(\Phi(w_{2}))\Leftrightarrow\Phi(w_{1})\perp^{Q}\Phi(w_{2}).$

Therefore $\Phi$ restricts to a locality linear bijection
$\Phi_{\top}:\mathrm{Sh}_{1,\top}(\mathbb{Q}\overline{U})\to\mathbb{Q}{\mathcal{F}}^{L}_{\top}$.
Finally, the multiplicativity for $\Phi$ in Eq. (41) restricts to one for
$\Phi_{\top}$. Hence $\Phi_{\top}$ is a locality algebra isomorphism.

(iii) This last assertion follows from Item (ii) and Corollary 4.5.(iii). ∎

We consider two special instances of maps $L:U\to{\mathcal{L}}_{\mathbb{Q}}$.

######  Example 4.10.

  1. (i)

Let $U$ be ${\mathbb{Z}}_{>0}$ equipped with the natural order and the
locality relation $n\top m\Leftrightarrow n\not=m.$ Define

$L:{\mathbb{Z}}_{>0}\to{\mathcal{L}}_{\mathbb{Q}},i\mapsto
z_{i},i\in{\mathbb{Z}}_{>0}.$

The corresponding set ${\mathcal{F}}^{L}_{\top}$ of ordered fractions is the
set ${\mathcal{F}}^{\rm Ch}$ of Chen fractions in Example 2.7.

  2. (ii)

Let $U$ be the set ${\mathcal{P}}_{\rm fin}({\mathbb{Z}}_{>0})$ of nonempty
finite subsets of ${\mathbb{Z}}_{>0}$. The order is the lexicographic order:
for elements

$I\coloneqq\\{i_{1}>i_{2}>\cdots>i_{r}\\},\quad
J\coloneqq\\{j_{1}>j_{2}>\cdots>j_{s}\\}$

in ${\mathcal{P}}_{\rm fin}({\mathbb{Z}}_{>0})$, define $I\geq J$ if either
the first nonzero element in the sequence

$i_{1}-j_{1},i_{2}-j_{2},\ldots,i_{\min\\{r,s\\}}-j_{\min\\{r,s\\}}$

is positive, or the above sequence of numbers are all zero and $r>s$. The
locality relation in ${\mathcal{P}}_{\rm fin}({\mathbb{Z}}_{>0})$ is:

$I\top J\Leftrightarrow I\cap J=\emptyset.$

Define

(45) $L:{\mathcal{P}}_{\rm
fin}({\mathbb{Z}}_{>0})\to{\mathcal{L}}_{\mathbb{Q}},\ \ I\mapsto
z_{I}\coloneqq\sum_{i\in I}z_{i},\quad\forall I\in{\mathcal{P}}_{\rm
fin}({\mathbb{Z}}_{>0}).$

The corresponding fractions
${\mathfrak{f}}^{L}\Big{(}\begin{array}[]{c}\scriptstyle{s_{1},\ldots,s_{k}}\\\
\scriptstyle{I_{1},\ldots,I_{k}}\end{array}\Big{)}$ are of the form

(46)
$\frac{1}{z_{I_{1}}^{s_{1}}(z_{I_{1}}+z_{I_{2}})^{s_{2}}\cdots(z_{I_{1}}+\cdots+z_{I_{k}})^{s_{k}}},\,\
s_{i}\in\mathbb{Z}_{>0},k\in\mathbb{N},I_{j}\cap
I_{j^{\prime}}=\emptyset\,{\rm if}\ j\neq j^{\prime}$

which we will call Speer fractions. Let ${\mathcal{F}}^{{\rm Sp}}$ denote the
set of Speer fractions and let

${\mathcal{M}}^{{\rm
Sp}}:={\mathcal{M}}_{{\mathbb{Q}}+}\left(\Pi^{Q}(\widehat{{\mathcal{F}}^{{\rm
Sp}}})\right).$

The term Speer fractions is to recognize that such fractions first appeared in
Speer’s work [Sp2] in the context of renormalisation. Notice that the linear
forms in a Speer fraction correspond to faces of a Chen cone. See Section 5
for details.

######  Remark 4.11.

We note that ${{\mathcal{M}}}_{\mathbb{Q}}^{\rm
Fe}\supset{{\mathcal{M}}}_{\mathbb{Q}}^{\rm Sp}$, yet whether they actually
differ is an open question.

As a direct consequence of Theorems 4.9, we obtain

######  Corollary 4.12.

The locality algebras ${\mathbb{Q}}{\mathcal{F}}^{\rm Ch}_{\top}$ and
${\mathbb{Q}}{\mathcal{F}}^{{\rm Sp}}_{\top}$ are locality polynomial algebras
on the set of fractions in ${\mathcal{F}}^{\rm Ch}$ and ${\mathcal{F}}^{{\rm
Sp}}$ respectively corresponding to the locality Lyndon words.

##  5\. Locality characters and analytic renormalisation

In this last section, as an application of our previous results, we address
Problems 1.1-1.3 raised in the introduction on Speer’s analytic
renormalisation. We first show that the pole structures of the generalised
Feynman amplitudes in Speer’s analytic renormalisation are of the form
introduced earlier. We then compare our constructions of locality generalised
evaluators with those of Speer, and obtain a transitivity group action on
locality generalised evaluators.

###  5.1. Speer’s $s$-families and Speer fractions

In his work on analytic renormalisation [Sp2, Sp3, Sp4] (see also [BR, DZ]),
Speer determined the possible pole structure of generalised Feynman amplitudes
(or regularised Feynman amplitudes). We show that these linear poles are
spanned by Speer fractions described in Eq. (46).

We first summarise Speer’s work, mostly following [Sp3]. A Feynman graph is
called $2$-connected if it cannot be disconnected by removing a vertex. A
family $\tt{E}$ of subgraphs of a Feynman graph $G$ is called a singularity
family or simply an $s$-family [Sp3, Definition 2] if

  1. (i)

every element in $\tt{E}$ is either 2-connected or a single line. Let
$\tt{E}^{\prime}$ denote the subset of 2-connected elements in $\tt{E}$;

  2. (ii)

$\tt{E}$ is nonoverlapping, that is, for $H_{1},H_{2}\in\tt{E}$, either
$H_{1}\subset H_{2}$ or $H_{2}\subset H_{1}$ or $H_{1}\cap H_{2}=\emptyset$;

  3. (iii)

no union of two or more disjoint elements of $\tt{E}$ is 2-connected;

  4. (iv)

$\tt{E}$ is maximal with these properties.

For a given Feynman graph $G$, the generalised Feynman amplitude
${\mathcal{T}}_{G}$ [Sp3, Definition 1]) is built from products of propagators
assigned to each edge $\ell$ of $G$, regularised by means of a complex number
$\lambda_{\ell}$. It enjoys a decomposition [Sp3, Formula (2.16)] as a sum
over all s-families $\tt{E}$ of $G$, of meromorphic functions
${\mathcal{T}}_{\tt{E}}$

${\mathcal{T}}_{G}=\sum_{\tt{E}}{\mathcal{T}}_{\tt{E}},$

resulting from writing the closed cone $\prod_{\ell\in
E(G)}\\{\alpha_{\ell}\geq 0\\}$ as a union $\cup_{\tt{E}}D(\tt{E})$ of closed
cones $D(\tt{E})$ associated with each s-family $\tt{E}$ of subgraphs of $G$.

For an s-family $\tt{E}$ of $G$, and $H$ in ${\tt{E}}$, let

$\Lambda(H)\coloneqq\sum_{\ell\in L(H)}(\lambda_{\ell}-1),$

where $L(H)$ is the set of edges of $H$. Let $\mu(H)$ be the superficial
divergence of $H$. According to Speer [Sp3] (see Eq. (2.21), Theorem 3 and the
remark that follows, see also [Sp4], Lemma 1.4 and its proof), the possible
poles of ${\mathcal{T}}_{\tt{E}}$ are simple poles given by

$\Lambda(H)-\frac{1}{2}\mu(H)=0,-1,-2,\ldots.$

Since we are only renormalising generalised Feynman amplitudes at
$\lambda_{\ell}=1,\ell\in L(G)$, the possible singularities we need to deal
with are of the form
$\bigg{(}\prod_{H\in\tt{E}^{\prime}}\Lambda(H)\bigg{)}^{-1}.$ By a change of
variables $z_{i}=\lambda_{\ell_{i}}-1$, with an ordering
$\ell_{1},\ldots,\ell_{|L(G)|}$ of $L(G)$, the $\Lambda(H)$ corresponds to the
linear form $z_{I(H)}$ in Eq. (45), for $I(H)=\\{i\,|\,\ell_{i}\in L(H)\\}$.
So we only need to deal with germs of the form

$\bigg{(}\prod_{H\in\tt{E}^{\prime}}z_{I(H)}\bigg{)}^{-1}\,h\vspace{-.2cm}$

with $h$ a holomorphic germ.

We now address Problem 1.1.

######  Proposition 5.1.

For any s-family $\tt{E}$ of $G$, the fraction

(47) $\bigg{(}\prod_{H\in\tt{E}^{\prime}}z_{I(H)}\bigg{)}^{-1}\vspace{-.1cm}$

lies in $\mathbb{Q}{\mathcal{F}}^{{\rm Sp}}$. Thus the germs of the
generalised Feynman amplitudes at $z=0$ are in ${\mathcal{M}}^{{\rm
Sp}}_{\mathbb{Q}}$.

###### Proof.

Since there is no overlaps between any two 2-connected subgraphs in an
s-family $\tt{E}$, the Hasse diagram of $\tt{E}^{\prime}$ is a rooted forest.
The flattening procedure used in [CGPZ2, Theorem 5.11] which involves the
flattening morphism defined in [CGPZ2, Definition 2.10], transforms this
rooted forest into a linear combination of ladder trees. Correspondingly, the
fraction in Eq. (47) is a linear combination of (ordered) Speer fractions. ∎

###  5.2. Generalised evaluators on locality subalgebras of meromorphic germs

Let ${\mathcal{A}}$ be a locality subalgebra of the algebra
${\mathcal{M}}_{\mathbb{Q}}$ equipped with the locality relation $\perp^{Q}$
of Definition 2.10.

######  Definition 5.2.

A locality generalised evaluator ${\mathcal{E}}$ on the locality algebra
$({\mathcal{A}},\perp^{Q})$ is a linear form
${\mathcal{E}}:{\mathcal{A}}\to\mathbb{C},$ such that

  1. (i)

${\mathcal{E}}(h)=h(0)$ for $h\in\mathcal{M}_{\mathbb{Q}+}$.

  2. (ii)

${\mathcal{E}}(f_{1}\cdot
f_{2})={\mathcal{E}}(f_{1})\cdot{\mathcal{E}}(f_{2})$ for
$f_{1},f_{2}\in{\mathcal{A}}$ with $f_{1}\perp^{Q}f_{2}$.

We use $E({\mathcal{A}})=E^{Q}({\mathcal{A}})$ to denote the set of locality
generalised evaluators on $({\mathcal{A}},\perp^{Q})$.

######  Remark 5.3.

Notice that a locality subalgebra
${\mathcal{A}}\subset{\mathcal{M}}_{\mathbb{Q}}$ is defined as a direct limit
${\mathcal{A}}=\underset{\longrightarrow}{\lim}{\mathcal{A}}_{k}$ with
${\mathcal{A}}_{k}\subset{\mathcal{M}}_{\mathbb{Q}}(\mathbb{C}^{k})$, so that
a linear form ${\mathcal{E}}:{\mathcal{A}}\to\mathbb{C}$ amounts to a family
of linear forms
${\mathcal{E}}_{k}:{\mathcal{A}}_{k}\to\mathbb{C},k\in\mathbb{N}$ such that

${\mathcal{E}}_{k}|_{{\mathcal{A}}_{k-1}}={\mathcal{E}}_{k-1},\quad k\geq 1.$

######  Example 5.4.

Let ${\rm ev}_{0}:{\mathcal{M}}_{\mathbb{Q}+}\to\mathbb{Q}$ be the evaluation
at $0$ defined as ${\rm ev}_{0}(h)=h(0)$. Using the map $\pi_{+}^{Q}$ defined
in Eq. (17), we build a locality generalised evaluator on ${\mathcal{A}}$ as
the composition

(48) ${{\mathcal{E}}^{Q}_{\rm MS}}\coloneqq{\rm ev}_{0}\circ\pi_{+}^{Q},$

which we call the locality minimal subtraction map since it is a
generalisation to higher dimensions of the one variable minimal subtraction
map obtained by projection onto the holomorphic part of the Laurent series
ring $\mathbb{C}[\varepsilon^{-1},\varepsilon]]$ followed by evaluation at
zero.

In his work [Sp1, Sp2, Sp3, Sp4], Speer achieved analytic renormalisation by
means of linear forms he called generalised evaluators. Let us compare these
generalised evaluators with those defined in Definition 5.2.

Let us first observe that, as in Example 2.7, a function is given in the
variables $z_{i}$ means that we have chosen a basis $\\{e_{i}\\}$ of the
space, and every element in the space is written in the form $\sum
z_{i}e_{i}$. In the presence of an inner product we choose $\\{e_{i}\\}$ to be
an orthonormal basis.

Following Speer (and see Example 3.9), define ${\mathcal{M}}^{\rm
Fe}(\mathbb{C}^{k})$ to be the spaces of meromorphic germs (at zero) $f$ with
the property that

$f:\mathbb{C}^{k}\to\mathbb{C}\ \text{ such that }\
f(z_{1},\ldots,z_{k})\cdot\prod_{I\subseteq[k]}z_{I}\,\text{ is holomorphic at
zero}.\vspace{-.2cm}$

Speer defines a generalised evaluator [Sp1] as a family of linear maps

${\mathcal{E}}\coloneqq\Big{\\{}{\mathcal{E}}_{k}:{\mathcal{M}}^{\rm
Fe}(\mathbb{C}^{k})\to\mathbb{C}\Big{\\}}_{k\in\mathbb{N}}$

satisfying the following conditions.

  1. (i)

(compatibility with the filtration) ${\mathcal{E}}_{k}|_{{\mathcal{M}}^{\rm
Fe}(\mathbb{C}^{k-1})}={\mathcal{E}}_{k-1},k\geq 1;$

  2. (ii)

(extension of ${\rm ev}_{0}$) ${\mathcal{E}}$ is the usual evaluation ${\rm
ev}_{0}$ at zero on holomorphic functions;

  3. (iii)

(partial multiplicativity) ${\mathcal{E}}(f_{1}\cdot
f_{2})={\mathcal{E}}(f_{1})\cdot{\mathcal{E}}(f_{2})$ if $f_{1}$ and $f_{2}$
depend on disjoint sets of variables $z_{i}$;

  4. (iv)

($\Sigma_{k}$-invariance) ${\mathcal{E}}$ is invariant under permutations of
the variables ${\mathcal{E}}_{k}\circ\sigma^{*}={\mathcal{E}}_{k}$ for any
$\sigma\in\Sigma_{k}$, with $\sigma^{*}f(z_{1},\ldots,z_{k})\coloneqq
f(z_{\sigma(1)},\ldots,z_{\sigma(k)})$;

There are also reality and continuity conditions which we do not discuss here.
Continuity of generalised evaluators requires enhancing the constructions
carried out here to a topological setting, which is the object of a joint work
[DPS] of the second author.

In practice, Speer builds such a generalised evaluator by setting

(49) ${\mathcal{E}}_{k}^{\rm
iter}\coloneqq\frac{1}{k!}\,\sum_{\sigma\in\Sigma_{k}}{\rm ev}^{{\rm
reg},z_{\sigma(1)}}_{0}\circ\cdots\circ{\rm ev}^{{\rm
reg},z_{\sigma(k)}}_{0},\vspace{-.2cm}$

where, for $1\leq i\leq k$, ${\rm ev}^{{\rm reg},z_{i}}_{0}(f)$ is defined by
${\rm ev}_{0}\circ\pi_{+}^{z_{i}}(f)$ when viewing $f$ as a meromorphic
function in the variable $z_{i}$, where $\pi^{z_{i}}_{+}$ is defined as in Eq.
(2).

######  Example 5.5.

We give some examples in the case of $k=2$.

  * a)

For $f(u,v)=\frac{u}{v}$, $g(u,v)=\left(\frac{u}{v}\right)^{2}$, we have
${\mathcal{E}}_{2}^{\rm iter}(f)={\mathcal{E}}_{2}^{\rm iter}(g)=0$;

  * b)

A change of variable $u=z_{1}-z_{2},v=z_{1}+z_{2}$ in $f$ and $g$ gives

$\tilde{f}(z_{1},z_{2})=\frac{z_{1}-z_{2}}{z_{1}+z_{2}},\quad\tilde{g}(z_{1},z_{2})=\left(\frac{z_{1}-z_{2}}{z_{1}+z_{2}}\right)^{2},\vspace{-.2cm}$

and we have ${\mathcal{E}}_{2}^{\rm iter}(\tilde{f})=0$ whereas
${\mathcal{E}}_{2}^{\rm iter}(\tilde{g})=1$.

We now show that our locality generalised evaluators are the generalised
evaluators à la Speer and that it provides a useful alternative to the
generalised evaluator ${\mathcal{E}}^{\rm iter}$ originally used by Speer.

######  Proposition 5.6.

Given any inner product $Q$, a locality generalised evaluator ${\mathcal{E}}$
in Definition 5.2 satisfies Conditions (i) – (iv) defining generalised
evaluators à la Speer.

###### Proof.

Indeed, conditions (i) follows from Remark 5.3, and (ii) is the same as
condition (i) in Definition 5.2. The locality multiplicative condition
(Definition 5.2.(ii)) for ${\mathcal{E}}$ is stronger than the partial
multiplicativity condition (iii) since disjointness of variables of two
functions implies the orthogonality for $Q$ when the variables correspond to
coordinates in an orthonormal basis for $Q$. Condition (iv) follows from
Theorem 5.1(ii) in [CGPZ3]. ∎

In order to address Problem 1.2, combining Corollary 5.8 and Proposition 5.6,
we observe that every generalised evaluator à la Speer factors through the
minimal subtraction scheme ${{\mathcal{E}}^{Q}_{\rm MS}}$.

We compare the two generalised evaluators ${\mathcal{E}}^{\rm iter}_{k}$ and
${{\mathcal{E}}^{Q}_{\rm MS}}$.

  1. (i)

We first observe a problem in the inductive procedure proposed by Speer since
the function $(z_{2},\cdots,z_{k})\mapsto{\rm ev}_{0}^{\rm
reg,z_{1}}(f(z_{1},\cdots,z_{k}))$ might be non-meromorphic, which is an
obstacle to implementing the composition ${\rm ev}_{0}^{\rm
reg,z_{j}}\circ{\rm ev}_{0}^{\rm reg,z_{1}}$ on $f$ for $j\neq 1$.

For example, the discontinuity at zero

${\rm ev}_{0}^{\rm
reg,z_{1}}(\tilde{f})=\left\\{\begin{array}[]{ll}1,&z_{2}=0,\\\
-1,&z_{2}\not=0,\end{array}\right.\,$

is an obstacle for the next step which gives the expression ${\rm
ev}_{0}^{{\rm reg},z_{2}}\circ{\rm ev}_{1}^{{\rm reg},z_{1}}(\tilde{f})$
arising in the definition ${\mathcal{E}}_{2}^{\rm iter}(\tilde{f})$. This also
suggests that there is no natural way to interpret Speer’s generalised
evaluator as a minimal subtraction scheme in multiple variables. In contrast,
our generalised evaluator ${{\mathcal{E}}^{Q}_{\rm MS}}$ is defined for
meromorphic germs with linear poles, so this does not pose a problem.

  2. (ii)

The partial multiplicativity property (ii) for Speer’s generalised evaluator
applies to a smaller set of pairs of functions than the one allowed by our
locality multiplicativity. For $\tilde{g}_{1}(z_{1},z_{2})=(z_{1}-z_{2})^{2}$
and $\tilde{g}_{2}(z_{1},z_{2})=(z_{1}+z_{2})^{-2}$ we have

${\mathcal{E}}_{2}^{\rm
iter}(\tilde{g}_{1}\,\tilde{g}_{2})\neq{\mathcal{E}}_{2}^{\rm
iter}(\tilde{g}_{1}){\mathcal{E}}_{2}^{\rm iter}(\tilde{g}_{2})\vspace{-.1cm}$

since ${\mathcal{E}}_{2}^{\rm iter}(\tilde{g}_{1}\,\tilde{g}_{2})=1$ and
${\mathcal{E}}_{2}^{\rm iter}(\tilde{g}_{1})={\mathcal{E}}_{2}^{\rm
iter}(\tilde{g}_{2})=0$. Yet the multiplicativity holds for
${{\mathcal{E}}^{Q}_{\rm MS}}$ since the linear forms $z_{1}-z_{2}$ and
$z_{1}+z_{2}$ are orthogonal (as before, the parameters $z_{i}$’s correspond
to coordinates in an orthonormal basis for $Q$) and we have

${{\mathcal{E}}^{Q}_{\rm
MS}}(\tilde{g}_{1}\,\tilde{g}_{2})=0={{\mathcal{E}}^{Q}_{\rm
MS}}(\tilde{g}_{1})\,{{\mathcal{E}}^{Q}_{\rm
MS}}(\tilde{g}_{2}).\vspace{-.1cm}$

  3. (iii)

As illustrated by Example 5.5, Speer’s generalised evaluator
${\mathcal{E}}^{\rm iter}_{k}$ depends on a choice of basis since
${\mathcal{E}}^{\rm iter}_{2}(g)\neq{\mathcal{E}}^{\rm iter}_{2}(\tilde{g})$.
It is not invariant even under an orthogonal transformation of the variables.
In contrast, our generalised evaluator ${{\mathcal{E}}^{Q}_{\rm MS}}$ does not
depend on such an orthogonal transformation. For the function $g$ in Example
5.5 with standard basis $z_{1},z_{2}$ under an inner product $Q$,
${{\mathcal{E}}^{Q}_{\rm MS}}(g)={{\mathcal{E}}^{Q}_{\rm MS}}(\tilde{g})=0$.

To conclude, the above observations speak in favour of the use of the global
minimal subtraction scheme ${{\mathcal{E}}^{Q}_{\rm MS}}$, since it is
globally defined on ${\mathcal{M}}_{\mathbb{Q}}^{\rm Fe}(\mathbb{C}^{k})$ and
is multiplicative on a large set of pairs.

###  5.3. Locality Galois actions on generalised evaluators

We consider the action of the locality Galois group on generalised evaluators.
Clearly, the group
$\mathrm{Gal}^{Q}({\mathcal{A}}/{\mathcal{M}}_{\mathbb{Q}+})$ acts on
$E^{Q}({\mathcal{A}})$:

(50)
$E^{Q}({\mathcal{A}})\times\mathrm{Gal}^{Q}({\mathcal{A}}/{\mathcal{M}}_{\mathbb{Q}+})\to
E^{Q}({\mathcal{A}}),\quad({\mathcal{E}},g)\mapsto{\mathcal{E}}\circ
g.\vspace{-.1cm}$

The subsequent theorem shows that the automorphism group ${\rm
Aut}^{Q}({\mathcal{A}})\cong\mathrm{Gal}^{Q}({\mathcal{A}}/{\mathcal{M}}_{\mathbb{Q}+})$
acts transitively on $E^{Q}({\mathcal{A}})$, and thus relates any locality
generalised evaluator to the locality minimal subtraction scheme
${{\mathcal{E}}^{Q}_{\rm MS}}$. In this respect,
$\mathrm{Gal}^{Q}({\mathcal{A}}/{\mathcal{M}}_{\mathbb{Q}+})$ can be regarded
as a renormalisation group on generalised evaluators.

######  Theorem 5.7.

For ${\mathcal{S}}\subseteq{\mathcal{F}}$, we consider the unitary locality
subalgebra
${\mathcal{A}}\coloneqq{\mathcal{M}}_{\mathbb{Q}+}^{Q}(\Pi^{Q}(\widehat{{\mathcal{S}}}))$
of ${\mathcal{M}}_{\mathbb{Q}}$. Suppose that
$\mathbb{Q}\Pi^{Q}(\widehat{{\mathcal{S}}})$ is a locality polynomial
subalgebra of $\widehat{{\mathcal{F}}}$ with a locality polynomial basis
$S\subset\Pi^{Q}({\mathcal{S}})$. Then every locality generalised evaluator
${\mathcal{E}}$ on ${\mathcal{A}}$ factorises through the minimimal
subtraction scheme ${{\mathcal{E}}^{Q}_{\rm MS}}$, that is, there is some
$\tilde{\varphi}$ in
$\mathrm{Gal}^{Q}({\mathcal{A}}/{\mathcal{M}}_{\mathbb{Q}+})$ such that

(51) ${{\mathcal{E}}^{Q}_{\rm MS}}\circ\tilde{\varphi}={\mathcal{E}}.$

###### Proof.

Let ${\mathcal{E}}\in E^{Q}({\mathcal{A}})$ be given. By assumption,
${\mathcal{B}}\coloneqq\mathbb{Q}\Pi^{Q}(\widehat{{\mathcal{S}}})$ is a
locality polynomial subalgebra of ${\mathcal{M}}_{\mathbb{Q}}$ with
$S\coloneqq\\{s_{\alpha}\\}\subset\Pi^{Q}({\mathcal{S}})$ as a set of locality
polynomial generators. Then the map

(52) $\varphi:S\to{\mathcal{B}},\quad s_{\alpha}\mapsto
s_{\alpha}+{\mathcal{E}}(s_{\alpha})$

extends to a locality algebra homomorphism $\varphi$ on ${\mathcal{B}}$.

For any $t$ in $\Pi^{Q}({\mathcal{S}})$, we now determine the form of
$\varphi(t)$. Write

$t=P(S)$

as a polynomial in $S$. Then $P$ is a sum of monomials of the form
$\Pi_{\alpha}s_{\alpha}$ with $s_{\alpha}$ all distinct due to the locality
algebraic independence. Further, let $W={\rm Supp}(t)$ and $N=\text{\rm
p-ord}(t)$. Then by Lemma 2.2, we can assume that all these monomials have
supporting space $W$ and p-order $N$. For each of these monomials, by the
locality multilplicativity, we have

$\varphi(\Pi_{\alpha}s_{\alpha})=\Pi_{\alpha}(s_{\alpha}+{\mathcal{E}}(s_{\alpha}))=\Pi_{\alpha}s_{\alpha}+{\rm
lower\ order\ terms}.$

Here the lower order terms come from $\Pi_{\alpha}s_{\alpha}$ by replacing one
or more factors $s_{\alpha}$ by ${\mathcal{E}}(s_{\alpha})$, so they indeed
have lower p-orders and smaller supporting spaces. Since all the monomials
$\Pi_{\alpha}s_{\alpha}$ have the same supporting space and p-order, we have

$\varphi(t)=\varphi(P(S))=P(S)+{\rm lower\ order\ terms}.$

The map $\varphi$ therefore defines an element in ${\rm Aut}_{{\rm
Res}}^{Q}({\mathcal{B}})$. The extension
$\tilde{\varphi}\in\mathrm{Gal}^{Q}({\mathcal{A}}/{\mathcal{M}}_{\mathbb{Q}+})$
of $\varphi$ obtained from Theorem 3.15 is a locality algebra homomorphism.
Further we have

${{\mathcal{E}}^{Q}_{\rm MS}}\circ\tilde{\varphi}(s_{\alpha})={\rm
ev}_{0}\circ\pi_{+}^{Q}\circ\tilde{\varphi}(s_{\alpha})={\rm
ev}_{0}\circ\pi_{+}^{Q}(s_{\alpha}+{\mathcal{E}}(s_{\alpha}))={\rm
ev}_{0}({\mathcal{E}}(s_{\alpha}))={\mathcal{E}}(s_{\alpha}).$

Hence, the locality algebra homomorphisms ${{\mathcal{E}}^{Q}_{\rm
MS}}\circ\tilde{\varphi}$ and ${\mathcal{E}}$ agree on the locality polynomial
generating set $S$ of ${\mathcal{A}}$. Therefore ${{\mathcal{E}}^{Q}_{\rm
MS}}\circ\tilde{\varphi}={\mathcal{E}}$ on ${\mathcal{B}}$. Since they also
agree on ${\mathcal{M}}_{\mathbb{Q}+}$, by Proposition 3.14, they agree on
${\mathcal{M}}_{\mathbb{Q}+}(\Pi^{Q}({\mathcal{S}}))$. ∎

As a direct consequence of Theorem 4.9 and Corollary 4.12, we obtain a
statement which addresses Problem 1.3

######  Corollary 5.8.

The space $E^{Q}({\mathcal{M}}^{\rm Ch}_{\mathbb{Q}})$ $\big{(}$resp.
$E^{Q}({\mathcal{M}}^{{\rm Sp}}_{\mathbb{Q}})$$\big{)}$ of locality
generalised evaluators on the locality algebra $\left({\mathcal{M}}^{{\rm
Ch}}_{\mathbb{Q}},\perp^{Q}\right)$ $\Big{(}$resp. $\left({\mathcal{M}}^{{{\rm
Sp}}}_{\mathbb{Q}},\perp^{Q}\right)\Big{)}$ is a homogeneous space of
$\mathrm{Gal}^{Q}\left({\mathcal{M}}^{{\rm
Ch}}_{\mathbb{Q}}/{\mathcal{M}}_{\mathbb{Q}+}\right)$ $\Big{(}$resp.
$\mathrm{Gal}^{Q}\left({\mathcal{M}}^{{{\rm
Sp}}}_{\mathbb{Q}}/{\mathcal{M}}_{\mathbb{Q}+}\right)$$\Big{)}$. In other
words, these groups act transitively on $E^{Q}({\mathcal{M}}^{\rm
Ch}_{\mathbb{Q}})$ $\big{(}$resp. $E^{Q}({\mathcal{M}}^{{\rm
Sp}}_{\mathbb{Q}})$$\big{)}$.

To finish the paper, we use multiple zeta values to build an example of
locality generalized evaluators and elements of the locality Galois group.

Recall that for $s_{1},\ldots,s_{k}$ in ${\mathbb{Z}}_{>0}$ with $s_{1}\geq
2$, the multiple zeta value (also called a multizeta value) at
$(s_{1},\cdots,s_{k})$ is

$\zeta\left(s_{1},\cdots,s_{k}\right):=\sum_{n_{k}>\cdots>n_{1}\geq
1}n_{1}^{-s_{1}}\cdots
n_{k}^{-s_{k}}=\prod_{i=1}^{k}\sum_{m_{i}=1}^{\infty}(m_{1}+\cdots+m_{i})^{-s_{i}}.$

For a locality Lyndon word $x_{0}^{s_{1}-1}x_{u_{1}}\ldots
x_{0}^{s_{k}-1}x_{u_{k}}\in\mathrm{Sh}_{1,\top}(\overline{U})$ and for the
corresponding “Lyndon Chen fraction”

${\mathfrak{f}}\Big{(}\begin{array}[]{c}\scriptstyle{s_{1},\ldots,s_{k}}\\\
\scriptstyle{{u_{1}},\ldots,{u_{k}}}\end{array}\Big{)}\coloneqq\frac{1}{z_{u_{1}}^{s_{1}}(z_{u_{1}}+z_{u_{2}})^{s_{2}}\cdots(z_{u_{1}}+z_{u_{2}}+\cdots+z_{u_{k}})^{s_{k}}}\,,\
u_{i},s_{i}\in\mathbb{Z}_{>0},k\in\mathbb{N},u_{i}\neq u_{j}\,{\rm if}\ i\neq
j,$

in ${\mathcal{F}}^{\rm Ch}_{\top}$ (see Example 4.10.(i)), define

(53)
${\mathcal{E}}^{\zeta}\left({\mathfrak{f}}\Big{(}\begin{array}[]{c}\scriptstyle{s_{1},\ldots,s_{k}}\\\
\scriptstyle{{u_{1}},\ldots,{u_{k}}}\end{array}\Big{)}\right):=\left\\{\begin{array}[]{ll}\zeta\left(s_{1},\cdots,s_{k}\right),&s_{1}\geq
2,\\\ 0,&s_{1}=1.\end{array}\right.$

Then by Corollary 4.12, this assignment extends to a unique locality algebra
homomorphism

${\mathcal{E}}^{\zeta}:{\mathcal{A}}^{\rm Chen}\to\mathbb{R}.$

Here $\mathbb{R}$ is equipped with the full locality condition
$\mathbb{R}\times\mathbb{R}$, implying the locality of the homomorphism
${\mathcal{E}}^{\zeta}$.

Note that the map on ${\mathcal{A}}^{\rm Chen}$ defined by Eq. (5.3) for _all_
locality Chen fractions is also a locality algebra homomorphism, following
[IKZ]. It therefore coincides with ${\mathcal{E}}^{\zeta}$. Hence we conclude
that assigning multiple zeta values to locality Chen fractions as in (53)
defines a locality generalized evaluator ${\mathcal{E}}^{\zeta}$. Then thanks
to Theorem 5.7, there is a transformation $\tilde{\varphi}$ in the locality
Galois group $\mathrm{Gal}^{Q}({\mathcal{A}}^{\rm
Chen}/{\mathcal{M}}_{\mathbb{Q}+})$, such that

${{\mathcal{E}}^{Q}_{\rm MS}}\circ\tilde{\varphi}={\mathcal{E}}^{\zeta}.$

Acknowledgments. The second author is grateful to the Perimeter Institute in
Waterloo where she was hosted on an Emmy Noether fellowship. This research is
supported by the National Natural Science Foundation of China (11890663 and
11821001).

Declaration of interests. The authors have no conflicts of interest to
disclose.

Data availability. Data sharing is not applicable to this article as no new
data were created or analyzed in this study.

## References

  * [BR] M. P. Bellon and E. I. Russo, Ward-Schwinger-Dyson equations in $\varphi^{3}_{6}$ quantum field theory, Lett. Math. Phys. 111 (2021), No. 42, 31 pp.
  * [BV1] N. Berline and M. Vergne, Local Euler-Maclaurin formula for polytopes, Mosc. Math. J. 7 (2007), 355-386.
  * [BV2] N. Berline and M. Vergne, The equivariant Todd genus of a complete toric variety, with Danilov condition, J. Algebra 313 (2007), 28-39.
  * [BP] N. Bogoliubov and O. S. Parasiuk. Über die Multiplikation der Kausalfunktionen in der Quantentheorie der Felder, Acta Math. 97 (1957), 227-266.
  * [B] F. Brown, Feynman amplitudes, coaction principle, and cosmic Galois group, Comm. Number Theory Phys. 11 (2017), 453-556.
  * [C] P. Cartier, A mad day’s work: from Gröthendieck to Connes and Kontsevich, Bull. Amer. Math. Soc. (N.S.) 38 (2001), 389-408.
  * [CFL] K.T. Chen, R.H. Fox and R.C. Lyndon, Free differential calculus, IV. The quotient groups of the lower central series, Ann. of Math. 68 (1958), 81-95.
  * [CGPZ1] P. Clavier, L. Guo, S. Paycha and B. Zhang, An algebraic formulation of the locality principle in renormalisation, Eur. J. Math. 5 (2019), 356-394.
  * [CGPZ2] P. Clavier, L. Guo, S. Paycha and B. Zhang, Renormalisation and locality: branched zeta values, IRMA Lect. Math. & Theor. Phys. 32 (2020), 85-132.
  * [CGPZ3] P. Clavier, L. Guo, S. Paycha and B. Zhang, Locality and renormalisation: universal properties and integrals on trees, J. Math. Phys. 61 (2020), 022301, 19 pp.
  * [CK1] A. Connes and D. Kreimer, Renormalisation in quantum field theory and the Riemann-Hilbert problem. I. The Hopf algebra structure of graphs and the main theorem, Comm. Math. Phys. 210 (2000), 249-273.
  * [CK2] A. Connes and D. Kreimer, Renormalisation in quantum field theory and the Riemann-Hilbert problem. II. The the $\beta$-function, diffeomorphisms and the renormalization group, Comm. Math. Phys. 216 (2001) 215-241.
  * [CM] A. Connes and M. Marcolli, Noncommutative Geometry, Quantum Fields and Motives, Amer. Math. Soc., 2019.
  * [DPS] R. Dahmen, S. Paycha and A. Schmeding, The space of meromorphic functions in several variables with linear poles: topological structure, Arxiv:2206.13993.
  * [DZ] N. V. Dang and B. Zhang, Renormalization of Feynman amplitudes on manifolds by spectral zeta regularization and blow-ups, J. Eur. Math. Soc. 23 (2021), 503-556.
  * [GPZ] L. Guo, S. Paycha and B. Zhang, Renormalisation and the Euler-Maclaurin formula on cones, Duke Math J., 166 (2017), 537-571.
  * [GPZ3] L. Guo, S. Paycha and B. Zhang, A conical approach to Laurent expansions for multivariate meromorphic germs with linear poles, Pacific J. Math 307 (2020), 159-196.
  * [GX] L. Guo and B. Xie, The shuffle relation of fractions from multiple zeta values, Ramanujan J. 25 (2011), 307-317.
  * [GZ] L. Guo and B. Zhang, Renormalization of multiple zeta values, J. Algebra 319 (2008), 3770-3809.
  * [Hep] K. Hepp, Proof of the Bogoliubov-Parasiuk theorem on renormalization, Comm. Math. Phys. 2 (1966), 301-326.
  * [t’H] G. ’t Hooft, Dimensional regularization and the renormalization group, Nucl. Phys. B61 (1973), 455-468.
  * [t’HV] G. ’t Hooft and M. Veltman, Regularization and renormalization of gauge fields, Nucl. Phys. B44 (1972), 189-213.
  * [IKZ] K. Ihara, M. Kaneko and D. Zagier, Derivation and double shuffle relations for multiple zeta values, Compositio Math 142 (2006), 307-338.
  * [MP] D. Manchon and S. Paycha, Nested sums of symbols and renormalized multiple zeta values, Int. Math. Res. Not. IMRN (2010), 4628-4697.
  * [P] J. E. Pommersheim, Toric varieties, lattice points and Dedekind sums, Math. Ann., 295, Issue: 1 (1993), 1-24.
  * [Ra] D. E. Radford, A natural ring basis for the shuffle algebra and an application to group schemes, J. Algebra 58 (1979), 432-454.
  * [Re] K. Rejzner, Locality and causality in perturbative algebraic quantum field theory, J. Math. Phys. 60 (2019), 122301.
  * [Sp1] E. Speer, Analytic renormalization, J. Math. Phys. 9 (1968), 1040.
  * [Sp2] E. Speer, On the structure of analytic renormalization, Comm. Math. Phys. 23 (1971), 23-36. Added note: Comm. Math. Phys. 25 (1972), 336.
  * [Sp3] E. Speer, Lectures on analytic renormalisation, Technical Report No. 73-067, 1972.
  * [Sp4] E. Speer, Analytic renormalization using many space-time dimensions, Comm. Math. Phys. 37 (1974), 83-92.
  * [Zh] J. Zhao, Multiple Zeta Functions, Multiple Polylogarithms and Their Special Values, World Scientific, 2016.
  * [Zi] W. Zimmermann, Convergence of Bogoliubov’s method of renormalization in momentum space, Comm. Math. Phys. 15 (1969), 208-234.