# A multidimensional generalization of Symanzik polynomials

Matthieu Piquerez<EMAIL_ADDRESS>CMLS, École
polytechnique, Université Paris-Saclay, 91128 Palaiseau Cedex, France

###### Abstract.

Symanzik polynomials are defined on Feynman graphs and they are used in
quantum field theory to compute Feynman amplitudes. They also appear in
mathematics from different perspective. For example, recent results show that
they allow to describe asymptotic limits of geometric quantities associated to
families of Riemann surfaces.

In this paper, we propose a generalization of Symanzik polynomials to the
setting of higher dimensional simplicial complexes and study their basic
properties and applications. We state a duality relation between these
generalized Symanzik polynomials and what we call Kirchhoff polynomials, which
have been introduced in recent generalizations of Kirchhoff’s matrix-tree
theorem to simplicial complexes. Moreover, we obtain geometric invariants
which compute interesting data on triangulable manifolds. As the name
indicates, these invariants do not depend on the chosen triangulation.

We furthermore prove a stability theorem concerning the ratio of Symanzik
polynomials which extends a stability theorem of Amini to higher dimensional
simplicial complexes. In order to show that theorem, we will make great use of
matroids, and provide a complete classification of the connected components of
the exchange graph of a matroid, a result we hope could be of independent
interest.

Finally, we give some idea on how to generalize Symanzik polynomials to the
setting of matroids over hyperfields defined recently by Baker and Bowler.

##  1\. Introduction

The aim of this paper is to introduce a family of polynomials that we call
_generalized Symanzik polynomials_ , and to study their geometric and
combinatorial properties.

Classical Symanzik polynomials arising in quantum field theory are associated
to _Feynman graphs_ and used for computing Feynman amplitudes. They are
defined as follows.

Let $G=(V,E)$ be a graph with vertex set $V$ and edge set $E$. Let
$\mathfrak{p}=(\mathfrak{p}_{v})_{v\in V}$ be a collection of vectors called
the _external momenta_ such that each $\mathfrak{p}_{v}$, the external
momentum of $v\in V$, is an element of the real vector space $\mathbb{R}^{D}$
for some positive integer $D$ and endowed $\mathbb{R}^{D}$ with a Minkowski
bilinear form. Moreover, we suppose that the collection of external momenta
verify the conservation of momenta hypothesis, namely that $\sum_{v\in
V}\mathfrak{p}_{v}=0$. Such a pair $(G,\mathfrak{p})$ is called a _Feynman
graph_. In what follows we will only consider the case $D=1$, but the results
can be extended to the more general setting as in [ABBF16].

The first Symanzik polynomial denoted by $\psi_{G}$ is defined by

(1) $\psi_{G}(x):=\sum_{T\in\operatorname{\mathcal{T}}}\prod_{e\not\in
T}x_{e},$

where $\operatorname{\mathcal{T}}$ denotes the set of spanning trees of $G$,
where $x=(x_{e})_{e\in E}$ is a collection of variables indexed by the edges
of the graph, and the product for a spanning tree
$T\in\operatorname{\mathcal{T}}$ is on all edges of $G$ which are not in $T$.
(A spanning tree of $G=(V,E)$ is a connected subgraph of $G$ which has vertex
set $V$ and which does not have any cycle.) Note in particular that the first
Symanzik does not depend on the external momenta.

The second Symanzik polynomial denoted $\phi_{G}$ is defined by

(2)
$\phi_{G}(\mathfrak{p},x):=\sum_{F\in\operatorname{\mathcal{SF}}_{2}}q(F)\prod_{e\not\in
F}x_{e}.$

In this formula, $\operatorname{\mathcal{SF}}_{2}$ denotes the set of spanning
forests of $G$ which have two connected components. (A spanning forest of $G$
is a subgraph with vertex set $V$ and without any cycle.) For a spanning
forest $F\in\operatorname{\mathcal{SF}}_{2}$, the term $q(F)$ is defined by
$q(F):=-\langle\mathfrak{p}_{F_{1}},\mathfrak{p}_{F_{2}}\rangle$ where $F_{1}$
and $F_{2}$ are the two connected components of $F$, and where, for
$i\in\\{1,2\\}$, $\mathfrak{p}_{F_{i}}$ is the sum of the momenta of vertices
in $F_{i}$. Using these polynomials, the Feynman amplitude can be computed
then as a path integral of $\exp(-i\phi_{G}/\psi_{G})$.

The above polynomials have many known interesting properties, most of them are
summarized in [BW10]. In this paper, we present natural generalizations of the
above definitions. In particular, we are going to extend the setting to higher
dimensional simplicial complexes. Other generalizations already exist, see
e.g. [GR07] and [KRTW10], though the purpose is different from ours. In
addition to introducing a larger family of polynomials with interesting
properties, we hope that this generalization will provide a better
understanding of these polynomials and will widens their applications.

Symanzik polynomials are known to have tight connection to other concepts in
mathematics. Before going to the heart of our construction and presenting an
overview of our results, we would like to to start by highlighting some of
these links and making a series of comments to motivate the study undertaken
in this paper.

First, Symanzik polynomials of graphs are known to be in a sense _dual_ to
Kirchhoff polynomials. These latter polynomials are those obtained applying
the well-known (weighted) Kirchhoff matrix-tree theorem, which counts the
number of spanning trees of a graph as the determinant of a matrix linked to
the Laplacian of the graph. Recently, the notion of a spanning tree has been
extended to the case of higher dimensional simplicial complexes, and
Kirchhoff’s theorem has been generalized to this setting, c.f. [DKM09, Lyo09]
and [BK16]. In order to generalize the definition of Symanzik polynomials to
this setting, it is thus natural to exploit and formulate an appropriate
version of the duality theorem in this setting. This is what we have done to
find our definition of generalized Symanzik polynomials. In the classical
case, this duality infers a determinantal formula for Symanzik polynomials.
This is still true after the generalization.

This leads us to our second point which concerns _determinantal probability
measures_ [Lyo03]. This tight relation between Kirchhoff polynomials, spanning
trees and determinants is studied thoroughly in the paper [Lyo09] of Lyons on
random complexes. In fact, what appears here are determinantal probabilities.
For instance, the probability for a uniform (or suitably weighted) spanning
tree of a graph to contain a specific edge can be easily computed because it
is given by the determinant of some matrix. The Laplacian of the graph is used
to define this matrix. In the same way, Kirchhoff polynomials are also
obtained using this Laplacian. By analogy, the matrix whose determinant gives
the first Symanzik polynomial allows us to compute the probability for a
spanning tree not to contain a specific edge. In this paper, we will also
generalize Kirchhoff and Symanzik polynomials in two different ways: first to
_spanning forests_ and second to _higher order determinants_ (the classical
definition is of order two). This naturally leads to _hyperdeterminants_ which
are also linked to determinantal probability measures as it was shown in
[EG09].

This brings us naturally to the third point, which is the link to  _the theory
of matroids_. Matroids are mathematical objects obtained by abstracting and
generalizing some of the main notions of linear algebra and, in particular, of
graph theory, as spanning trees for example (we refer to [Oxl11] for further
information). In [BW10], the link between matroid theory and Symanzik
polynomials is noticed. We go further than [BW10] by defining Symanzik
polynomials for matroids and for their recent generalization _matroids over
hyperfields_ obtained by Baker and Bowler in [BB16]. Apart from this
generalization, we show that matroids play a major role in understanding some
interesting features of the Symanzik polynomials. These purely combinatorial
results about matroids, which we hope should be of independent interest,
concerns the exchange properties between _independent sets_ of a general
matroid, which generalize well-known exchange properties between bases. For a
connected graph, it is well-known that any spanning tree can be obtained from
any other by a sequence consisting of exchanging one edge at a time. The
exchange properties between spanning trees and spanning 2-forests in a graph
were studied by Amini in [Ami16] for the purpose of understanding the
asymptotic of some geometrically defined quantities in a family of Riemann
surfaces. Generalizing the definition of the exchange graph of a graph from
[Ami16] to any general matroid, we provide a complete characterization of the
connected components of this graph and use it to generalize the _stability
theorem_ of [Ami16] to higher dimensional simplicial complexes.

Let us make two further remarks at this point. It is well-known that the
connectivity properties of different kind of exchange graphs is the first step
for creating simple and efficient random generation of bases based on Markov
chains, c.f. [FM92] and the recent work of [ALOV18] based on combinatorial
Hodge theory for matroids [AHK18]. What we prove in this paper is another
geometric manifestation of the exchange properties in matroids. Furthermore,
the results we obtain on the exchange graph of a matroid has very close ties
to an old conjecture of White on toric ideals of matroids [Whi80, Bla08,
LM14]. In fact, by the work of [Bla08], a part of White’s conjecture can be
rephrased as whether a particular generalization of our exchange graph is
always connected, c.f. Remark 4.17. So our results and methods might lead to a
new approach in that direction.

As we mentioned previously, Symanzik polynomials for graphs originated from
Physics, and recent advances has shown that their properties govern the
mathematics of Feynman amplitudes, see e.g. [BB03, BEK06, Bro15, BS12, BY11].
In particular, Brown shows in [Bro15] how certain arithmetic factorization
properties satisfied by the Symanzik polynomials of graphs and their minors
play a crucial role in defining the _cosmic Galois group_ , a Galois group for
Feynman amplitudes.

From a point of view inclined towards Physics, we will show that our
generalized polynomials present a certain stability (rather different from the
stability theorem evoked above). This means that if we start with a
sufficiently nice topological space $\mathcal{S}$, then the Symanzik
polynomials of any triangulation of $\mathcal{S}$ come from (more precisely
can be factorized into) the same Symanzik polynomial, which can be directly
defined then as the Symanzik polynomial of $\mathcal{S}$. In other words, this
stability indicates that these polynomials might be useful for computing
interesting features of continuous objects by approximating them with discrete
ones. For instance, path integrals are fundamental in the formulation of
quantum field theory. But they are generally ill-defined. A way to go through
this difficulty is to discretize the paths, and Regge calculus gives a way to
do this (originally in the framework of quantum gravity), see e.g. the
original article of Regge [Reg61], the introductory paper [Wil92], and to a
more recent approach [Ori07] in the framework of group field theory. We have
found striking similarities between the geometry of generalized Symanzik
polynomials and Regge calculus, although at this point we are not still able
to formalize this.

About the geometrical data computed by generalized Symanzik polynomials, in
simple cases as compact orientable manifolds endowed with a volume form, the
Symanzik polynomial simply compute the volume of the manifold (setting the
good values to variables). There is not always a simple interpretation for the
value computed in more complex cases. However, for general metric graphs, the
first Symanzik polynomial evaluated at positive reals computes the volume of
the Jacobian torus of the graph, which is defined in [KS00].

The Symanzik polynomials of graphs has been generalized in the context of non-
commutative quantum field theory [KRTW10]. The article [KRV11] shows how these
new polynomial invariants derive from the Bollobás-Riordan polynomial, which
is a generalization of the Tutte polynomial for ribbon graphs. (Classical
Symanzik polynomials are in fact particular multivariate versions of the Tutte
polynomials [Sok05], (4.11).) The Bollobás-Riordan polynomial itself has been
generalized to higher dimensions in [AGH13] and [Avo16]. A recent paper
[KMT18] provides a generalization of these polynomials in the context of Hopf
algebras. Another recent paper [BC17] studies this kind of generalization for
a different purpose. It concerns the geometric Brownian motion, also linked to
quantum theory. As far as we know, multidimensional generalizations of
Symanzik polynomials have not been studied prior to this work, and it would be
certainly valuable to compare our approach to the above cited works.

Symanzik polynomials of graphs form a family of graph polynomials with very
interesting properties. They are specializations of the multivariate Tutte
polynomial. Recent papers study graph polynomials which usually reveal
surprising geometric properties. It is not hard to prove that the first
Symanzik polynomials of the connected graphs verify the half-plane property,
i.e., they are nonvanishing whenever all the variables lie in the open right
complex half-plane. The papers [COSW04], [Brä07] and [BBL07] study invariant
polynomials verifying this property and other linked properties. They might be
extended in some cases to our generalization.

Moreover, in [Ami16], it has been proved that a bounded geometrical
deformation of the graph only induces a variation of the ratio of the two
Symanzik polynomials that is bounded independently of the initial geometry. At
the end of this article, we will generalize this (previously mentioned)
stability theorem. The initial goal of Amini in [Ami16] was to give a
combinatorial proof of a result he obtained with Bloch, Burgos Gil and Fresán
in [ABBF16]. In that work, they expressed the asymptotic of the Archimedean
height pairing between two $0$-divisors on degenerating families of Riemann
surfaces as to the ratio of the two Symanzik polynomials. The generalization
presented in this paper should have links to the generalization of that work
to higher dimensional varieties. In another direction, Scott and Sokal study
in [SS14] strict monotonicity properties of the inverse powers of the
Kirchhoff polynomials of graphs and matroids. They give a complete
characterization of the exponents in the strict monotonicity range for series-
parallel graphs, and raise several questions and conjectures. One might
naturally wonder if such results could be generalized to the setting of
generalized Symanzik polynomials studied in this paper.

In the rest of this introduction, we give a quick overview of the main results
of this paper.

###  1.1. The generalization

The idea of the generalization comes from Kirchhoff’s matrix-tree theorem
(observed by Kirchhoff [Kir47]) in its weighted form. We briefly sketch the
idea here. Example 2.8 contains all the details. Let $G=(V,E)$ be a connected
graph of vertex set $V$ and edge set $E=\\{e_{1},\dots,e_{n}\\}$. Set a weight
$y_{e}\in\mathbb{R}$ on each edge $e\in E$. Set $Q$ the incident matrix of
$G$, and $Y$ the diagonal matrix
$\operatorname{diag}(y_{e_{1}},\dots,y_{e_{n}})$. Then, the theorem states
that

(3) $\sum_{T\in\operatorname{\mathcal{T}}}\prod_{e\in
T}y_{e}=\operatorname{det}(\widetilde{Q}Y{\widetilde{Q}}^{{}^{\intercal}}),$

where we removed a row from $Q$ to get $\widetilde{Q}$. Let
$E^{\prime}\subseteq E$ be a subset of $|V|-1$ edges. We restrict the matrix
$\widetilde{Q}$ to the columns indexed by the elements of $E^{\prime}$. We
denote this square matrix by $\widetilde{Q}_{E^{\prime}}$. A step of the proof
is to notice that $\operatorname{det}(\widetilde{Q}_{E^{\prime}})^{2}$ equals
$1$ if $E^{\prime}$ corresponds to a spanning tree of $G$, $0$ otherwise.
Thus, the first Symanzik polynomial verifies

$\psi_{G}(x)=\sum_{\begin{subarray}{c}E^{\prime}\subseteq E\\\
|E^{\prime}|=|V|-1\end{subarray}}\operatorname{det}(\widetilde{Q}_{E^{\prime}})^{2}\prod_{e\not\in
E^{\prime}}x_{e}.$

In the right-hand member, the graph $G$ is almost absent. The main data we use
is its incident matrix. Thus, one can define the Symanzik polynomial of any
matrix.

However, we needed to delete a row from the matrix. When the matrix does not
come from graphs, we could have to remove several rows, and the result could
depend on the chosen rows. To avoid any problem, we introduce a very useful
tool: the standard inner product on the exterior algebra. This is used in
[Lyo03] by Lyons to deal with a subject not far from ours, as explained above.

#### Notations and conventions

If $p,q$ are integers with $p\leqslant q$, then the set $\\{p,p+1,\dots,q\\}$
will be written $[p\,\raisebox{0.86108pt}{.}\,\raisebox{0.86108pt}{.}\,q]$, or
more simply $[q]$ if $p=1$. Let $I$ be a finite set. Then $|I|$ is its
cardinality and $\operatorname{\mathcal{P}}(I)$ is its power set. If
$J\in\operatorname{\mathcal{P}}(I)$ is a subset of $I$ and if there is no
ambiguity, then $J^{c}:=I\setminus J$ denotes the complement of $J$. If $i\in
I^{c}$, then $I+i:=I\cup\\{i\\}$, and if $i\in I$, then
$I-i:=I\setminus\\{i\\}$ (using these notations means, respectively, that
$i\in I^{c}$ and that $i\in I$). If there is an ordering on the elements $I$,
we will also use $I$ to denote the ascending family of elements in $I$ (see
below).

Let $n$ be a positive integer. In the whole article, $x=(x_{1},\dots,x_{n})$
will be a family of variables. $\mathbb{Z}[x]$ is the ring of polynomials over
$\mathbb{Z}$ with variables $x_{1},\dots,x_{n}$. Following a usual notation,
if $I\subseteq[n]$,

$x^{I}:=\prod_{i\in I}x_{i}.$

If $\Delta$ is a set, $\mathbb{Z}\langle\Delta\rangle$ is the free
$\mathbb{Z}$-module on $\Delta$. By convention, if $a\in\mathbb{Z}$, then

$a^{0}=\left\\{\begin{array}[]{ll}0&\text{if $a=0$,}\\\
1&\text{otherwise.}\end{array}\right.$

Let $n,p$ be positive integers. Then $\mathcal{M}_{p,n}(\mathbb{Z})$ will
denote the set of matrices with $p$ rows and $n$ columns over $\mathbb{Z}$ and
$\mathcal{M}_{n}(\mathbb{Z})$ is a simpler notation for
$\mathcal{M}_{n,n}(\mathbb{Z})$. If $P\in\mathcal{M}_{n,p}(\mathbb{Z})$, then
$P^{{}^{\intercal}}\in\mathcal{M}_{p,n}(\mathbb{Z})$ denotes the transpose of
$P$.

Let $n,p,q$ be positive integers. Let $u=(u_{1},\dots,u_{n})$ be a family of
$n$ vectors in $\mathbb{R}^{p}$. Then, when there is no ambiguity, the same
letter in uppercase, $U$, will denote the matrix in
$\mathcal{M}_{p,n}(\mathbb{R})$ whose columns are the coordinates of the
vectors of $u$. Reciprocally, if $U$ is a matrix, then one can associate to
$U$ a family of vectors $u$. If $I=(i_{1},\dots,i_{q})$ is a family of
elements of $[n]$, then $u_{I}$ will be the family
$(u_{i_{1}},\dots,u_{i_{q}})$, and $U_{I}\in\mathcal{M}_{p,q}(\mathbb{R})$
will be the associated matrix. Moreover, $u^{{}^{\intercal}}$ will be the
family of $p$ vectors in $\mathbb{R}^{n}$ associated to $U^{{}^{\intercal}}$.
If $I$ is an (unordered) subset of $[n]$ of size $q$, then $u_{I}$, resp.
$U_{I}$, will denote the family $u_{(i_{1},\dots,i_{q})}$, resp. the matrix
$U_{(i_{1},\dots,i_{q})}$, where $I=\\{i_{1},\dots,i_{q}\\}$ and
$i_{1}<i_{2}<\dots<i_{q}$.

If $u$ and $v$ are two families of elements of a same set of respective size
$n$ and $m$, then $u\star v$ will denote the concatenation of the two
families, i.e.,

$u\star v=(u_{1},\dots,u_{n},v_{1},\dots,v_{m}).$

By extension, if $U\in\mathcal{M}_{p,n}(\mathbb{R})$ and
$V\in\mathcal{M}_{p,m}(\mathbb{R})$ are two matrices, then $U\star V$ is the
matrix of $\mathcal{M}_{p,n+m}(\mathbb{R})$ associated to $u\star v$.

Let $\operatorname{\bigwedge}\mathbb{R}^{p}=\bigoplus_{l\geqslant
0}\operatorname{\bigwedge}^{l}\mathbb{R}^{p}$ be the exterior algebra over
$\mathbb{R}^{p}$. With the notations of the previous paragraphs, we will
denote by $\mathbf{u}\in\operatorname{\bigwedge}^{n}\mathbb{R}^{p}$, resp.
$\mathbf{u}_{I}\in\operatorname{\bigwedge}^{q}\mathbb{R}^{p}$, the exterior
product $u_{1}\wedge\dots\wedge u_{l}$, resp. $u_{i_{1}}\wedge\dots\wedge
u_{i_{q}}$. We endow $\operatorname{\bigwedge}\mathbb{R}^{p}$ with the
standard inner product: if $n$ and $m$ are two nonnegative integers, if $u$,
$v$ are two families of respective sizes $n$ and $m$ in $\mathbb{R}^{p}$, then

$(\mathbf{u},\mathbf{v})=\begin{cases}\operatorname{det}(U^{{}^{\intercal}}V)&\text{if
$n=m$,}\\\ 0&\text{otherwise.}\end{cases}$

We have the associated norm
$\lVert\mathbf{u}\rVert:=(\mathbf{u},\mathbf{u})^{1/2}$.

Note that, if $e$ is the standard basis of $\mathbb{R}^{p}$, then the family
$(\mathbf{e}_{I})_{I\subseteq[p],|I|=l}$ forms an orthonormal basis of
$\operatorname{\bigwedge}^{l}\mathbb{R}^{p}$.

We have the standard inclusion $\mathbb{Z}^{p}\subset\mathbb{R}^{p}$. If $F$
is a vector subspace of $\mathbb{R}^{p}$, then we denote by $F_{\mathbb{Z}}$
the sublattice $F\cap\mathbb{Z}^{p}$. If $u$ is a family of vectors in
$\mathbb{Z}^{p}\subset\mathbb{R}^{p}$, then we denote by $\mathbb{Z}u$ the
sub-$\mathbb{Z}$-module generated by $u$, and by $\langle u\rangle$ the
subspace of $\mathbb{R}^{p}$ spanned by $u$. If $A$ is a
sub-$\mathbb{Z}$-module of $\mathbb{Z}^{p}$, we will denote by $\lVert
A\rVert$ the covolume of $A$, i.e., the value $\lVert\mathbf{f}\rVert$ where
$f$ is any basis of $A$.

If $U\in\mathcal{M}_{p,n}(\mathbb{Z})$, then $\operatorname{Im}(U)$, resp.
$\ker(U)$, will always denote vector subspaces, i.e., the image, resp. kernel,
of $U$ in $\mathbb{R}^{p}$, resp. $\mathbb{R}^{n}$.

Let $n$ and $p$ be two positive integers. Let $u$ be a family of $n$ vectors
of $\mathbb{Z}^{p}\subset\mathbb{R}^{p}$. Let $r$ be the rank of $u$. We
define the Symanzik polynomial of $u$ by

$\operatorname{Sym}_{2}(u;x):=\frac{1}{{\lVert\langle
u\rangle_{\mathbb{Z}}\rVert}^{2}}\sum_{\begin{subarray}{c}I\subseteq[n]\\\
|I|=r\end{subarray}}\lVert\mathbf{u}_{I}\rVert^{2}x^{I^{c}}.$

Actually, one can replace the exponents $2$ by any nonnegative even integer
$k$. We will call $k$ the _order_ of the polynomial (see Definition 2.9). The
order $0$ has some theoretical interests, as we will see for matroids.
Moreover, Theorem 5.1 about stability can be generalized without great effort
for higher orders. That is why, we will study the case of all the orders in
this article. This widens the set of interesting polynomials. However, in this
introduction, we set $k=2$.

The main result of Section 2 is the close link between Symanzik polynomials
and the Kirchhoff theorem. To enlight this link, we introduce the Kirchhoff
polynomial of $u$:

$\operatorname{Kir}_{2}(u;x):=\frac{1}{{\lVert\langle
u\rangle_{\mathbb{Z}}\rVert}^{2}}\sum_{\begin{subarray}{c}I\subseteq[n]\\\
|I|=r\end{subarray}}\lVert\mathbf{u}_{I}\rVert^{2}x^{I}.$

Then, the link is given by the duality theorem (Theorem 2.12). If $v$ is a
family of maximal rank of $\ker(U)_{\mathbb{Z}}$, then the theorem states that
there exists an explicit factor $c\in\mathbb{Q}$ such that

$\operatorname{Kir}_{2}(v^{{}^{\intercal}};x)=c\operatorname{Sym}_{2}(u;x).$

Here is the generalization of the second Symanzik polynomial in a specific
case (see Definition 2.16 for the general case). If $a$ is an element of
$\mathbb{Z}u$, then the Symanzik polynomial of $u$ with parameter $a$ is

$\operatorname{Sym}_{2}(u;(a);x):=\operatorname{Sym}_{2}(u\star(a);x\star(0)).$

The justification of the definitions of Symanzik polynomials without
parameters and of Kirchhoff polynomials are given respectively in Examples
2.11 and 2.8. However, the link between Symanzik polynomials with parameters
and second Symanzik polynomials is harder to establish. We need Proposition
2.19. In this proposition, some orientations hidden in Symanzik polynomials
with parameters appear. These orientations are already well-known. They are
called _chirotopes_ in the theory of oriented matroid (see [FL78], [GR13]).
One can think about them in the following way. We take a spanning $2$-forests
$(F_{1},F_{2})$ of a graph. Then we can orient all edges that link $F_{1}$ and
$F_{2}$, from $F_{1}$ to $F_{2}$. Much detail is given in Example 2.20.

Propositions 2.5, 2.15 and 2.18 state determinantal formulæ similar to that of
(3). For instance, let $v$ be a basis of $\ker(U)$ belonging to
$\mathbb{Z}^{n}$, $a\in\mathbb{Z}u$ and $b\in\mathbb{Z}^{n}$ such that $Ub=a$.
Set $\widetilde{v}:=v\star(b)$. Then,

$\operatorname{Sym}_{2}(u;(a);x)=c\operatorname{det}(\widetilde{V}^{{}^{\intercal}}X\widetilde{V}),$

for some factor $c$ we explicit in Proposition 2.15, where
$X=\operatorname{diag}(x_{1},\dots,x_{n})$.

###  1.2. Geometrical aspects of Symanzik polynomials

Following an idea of Kalai in [Kal83], Duval, Klivans and Martin stated in
[DKM09] a generalization of Kirchhoff’s theorem for simplicial complexes. This
theorem counts the number of generalized spanning trees with some weights. In
Subsection 3.1, we will call these spanning trees _0-forests_ (or, more
exactly, we use the more convenient definition of spanning trees of Bernardi
and Klivans in [BK16]). Readers who are not familiar with simplicial homology
theory could consult the details in Subsection 3.1.

The definition of a $0$-forest is really simple: it is a maximal simplicial
subcomplex without nontrivial cycles of maximal dimension. To be more precise,
let $\Delta$ be a finite simplicial complex of dimension $d$, and let
$\partial_{\Delta}$ be the $d$-th boundary operator of $\Delta$. Let $\Gamma$
be a subcomplex of $\Delta$ with the same $(d-1)$-skeleton, and let
$\partial_{\Gamma}$ be the $d$-th boundary operator of $\Gamma$. Then, for an
integer $\kappa$, $\Gamma$ is a _$\kappa$ -forest_ of $\Delta$ if
$\ker(\partial_{\Gamma})=\\{0\\}$ and
$\operatorname{rk}(\partial_{\Delta})-\operatorname{rk}(\partial_{\Gamma})=\kappa$.
Forests in higher dimensions mostly behave like forests in graphs. Moreover,
the two notions coincide on connected graphs. In this case, $\kappa$-forests
are the spanning forests with $\kappa+1$ connected components.

The boundary operator $\partial_{\Delta}$ is a linear map. Let $U$ be the
associated matrix for some natural bases of simplicial $d$-chains and of
simplicial $(d-1)$-chains. It is natural to define the Symanzik polynomial,
resp. Kirchhoff polynomial, of $\Delta$ as the one of $u$. Then
$\operatorname{Kir}_{2}(\Delta;x)$ is what we get applying the higher
dimensional Kirchhoff weighted theorem to $\Delta$ (see Theorem 3.7):

$\operatorname{Kir}_{2}(\Delta;x)=|\operatorname{Tor}(\operatorname{H}_{d-1}(\Delta))|^{2}\sum_{\text{$\Gamma$
$0$-forest of
$\Delta$}}|\operatorname{B}_{d-1}(\Delta)\big{/}\operatorname{B}_{d-1}(\Gamma)|^{2}\prod_{\delta\text{
$d$-face of }\Gamma}x_{\delta},$

where $\operatorname{B}_{d-1}(\Delta)$, resp.
$\operatorname{B}_{d-1}(\Gamma)$, is the group of $(d-1)$-boundaries of
$\Delta$, resp. of $\Gamma$, and
$\operatorname{Tor}(\operatorname{H}_{d-1}(\Delta))$ denotes the torsion part
of the $(d-1)$-th homology group of $\Delta$.

Unlike the case of graphs, setting $x=(1,\dots,1)$ does not give the number of
$0$-forests. Finding this number is difficult in general: there exist examples
where all the coefficients are not equal (look at Example 3.27). Knowing the
value of the Kirchhoff polynomial for all order $k$ might help us.

However, the weighted number of $0$-forests for the order $k=2$ seems to be
the most natural object to consider. The first reason is in [BK16]. The
authors explain that this number counts $0$-forests of $\Delta$ taking into
account a _fitted orientation_. The second reason is that Kalai computed in
[Kal83] the weighted number of $0$-forests of the complete $d$-dimensional
simplicial complex over $N$ vertices. He obtains a pretty nice formula:
$N^{\binom{N-2}{d}}$.

Now, for each facet $\delta$ of $\Delta$, assign to the variable $x_{\delta}$
a positive number $y_{\delta}\in\mathbb{R}_{+}$ that represents the volume of
$\delta$. This is natural. For example, let $\mathcal{S}$ be a compact
oriented manifold of dimension $d$, and let $\omega$ be a volume form on it.
Let $\Delta$ be a triangulation of $\mathcal{S}$ (see Subsection 3.4 for a
rigorous definition of triangulation). Then $\omega$ induces a volume
$y_{\delta}\in\mathbb{R}_{+}$ for each facet $\delta$ of $\Delta$. It is not
hard to verify that

$\operatorname{Sym}_{2}(\Delta;y)=\operatorname{Vol}(\mathcal{S}).$

This is more general. Set $\mathcal{S}$ the topological space of any
$d$-dimensional finite CW-complex. We endow $\mathcal{S}$ with a diffuse
measure $\mu$ (i.e., a measure which is $0$ on any subset homeomorphic to
$[0;1]^{d-1}$). Let $\Delta$ be a triangulation of $\mathcal{S}$, and
$y=(y_{\delta})$ be the induced volume for each facet $\delta$. Then

$\operatorname{Sym}_{2}(\Delta;y)\text{ does not depend on the chosen
triangulation $\Delta$.}$

The proof essentially consists on defining directly
$\operatorname{Sym}_{2}(\mathcal{S};\mu)$, which will be the value we get for
any triangulation.

The case of a metric graph $\mathcal{G}$ is of particular interest.
$\operatorname{Sym}_{2}(\mathcal{G};\mu)$ (with $\mu$ induced by the metric)
is the volume of the Jacobian torus of $\mathcal{G}$ introduced by Kotani and
Sunada in [KS00] (see Example 3.12).

###  1.3. Stability theorem

Let $v$ be any free family of $n$ elements of $\mathbb{R}^{p}$. Let $a$ be an
element of $\mathbb{R}^{p}$ independent from $v$ and let $w=v\star(a)$. Let
$y(t)=(y_{1}(t),\dots,y_{n}(t))$ be a family of $n$ elements of
$\mathbb{R}_{+}$ depending on some parameter $t$. Let $Y(t)$ be the diagonal
matrix $\operatorname{diag}(y_{1}(t),\dots,y_{n}(t))$. Let $Z(t)$ be an
element of $\mathcal{M}_{n}(\mathbb{R})$ depending on $t$ whose entries are
bounded. Theorem 5.1 states, with different notations, that there exists two
constant $c,C\in\mathbb{R}_{+}$ such that, for any $t$ verifying
$y_{1}(t),\dots,y_{n}(t)\geqslant C$,

(4)
$\left|\frac{\operatorname{det}(W^{{}^{\intercal}}Y(t)W)}{\operatorname{det}(V^{{}^{\intercal}}Y(t)V)}-\frac{\operatorname{det}(W^{{}^{\intercal}}(Y(t)+Z(t))W)}{\operatorname{det}(V^{{}^{\intercal}}(Y(t)+Z(t))V)}\right|\leqslant
c.$

This result generalizes Theorem 1.1 of [Ami16]. Notice that both ratios are of
degree $1$ in $y$. Moreover, even if the ratios were of degree zero, terms
like $y_{1}/y_{2}$ are unbounded at infinity. Thus this property is
sufficiently rare among ratios of polynomials to be noticed.

Let us roughly explain what this theorem implies for Symanzik polynomials. For
the sake of simplicity, we assume that $Z$ is the diagonal matrix
$\operatorname{diag}(z_{1}(t),\dots,z_{n}(t))$ for all $t$. One can think
about the left-hand member of (4) as the absolute value of

$\frac{\operatorname{Sym}_{2}(\Delta;(b);y)}{\operatorname{Sym}_{2}(\Delta;y)}-\frac{\operatorname{Sym}_{2}(\Delta;(b);y+z)}{\operatorname{Sym}_{2}(\Delta;y+z)},$

for some simplicial complex $\Delta$ and some parameter $b$ (which is a
$(d-1)$-boundary of $\Delta$). Thus, the first ratio simply corresponds to the
ratio of the two Symanzik polynomials. The second one corresponds to the same
ratio after a bounded deformation of the geometry of $\Delta$. Hence, such a
deformation only induces a variation of the ratio of the two Symanzik
polynomials that is bounded independently of how great are the volumes of the
facets.

###  1.4. Structure of the paper

Section 2 is devoted to the generalization in an abstract setting, and to the
important duality theorem (Theorem 2.12) linking Symanzik polynomials to the
well-known Kirchhoff theorem. In the third section we develop applications of
Symanzik polynomials to geometry. The two last sections follow Sections 2 and
3 of [Ami16]. Section 4 describes a combinatorial result about what we call
the exchange graph of a matroid, which is needed for the last section. In this
last section, we state and prove the stability theorem.

### Acknowledgements

We whish to express our thanks to our Ph.D. supervisor Omid Amini for
suggesting the problem, for his many advices and for careful and patient
reading and correcting of the paper. We gratefully acknowledge the many
references given by Adrien Kassel, Vincent Rivasseau and Alan Sokal, and the
helpful suggestions of Spencer Bloch and Razvan Gurau.

##  2\. Kirchhoff and Symanzik polynomials and duality

###  2.1. Preliminaries

We begin this section setting up some more tools in order to deal with
Symanzik polynomials of higher orders.

In the whole article, $k$ will always be any _nonnegative even_ integer.

Let $p$ be a positive integer. If $k$ is nonzero, let
$a_{1}=(a_{1,1},\dots,a_{1,p}),\dots,a_{k}$ be $k$ vectors of
$\mathbb{R}^{p}$. We define the _standard $k$-multilinear symmetric product_
to be

$(a_{1},\dots,a_{k})_{k}:=\sum_{i=1}^{p}\prod_{j=1}^{k}a_{j,i}.$

If $y$ is a family of $p$ variables, then we define the _$k$ -multilinear
symmetric product associated to $y$_ by

$(a_{1},\dots,a_{k})_{k,y}:=\sum_{i=1}^{p}\bigg{(}\prod_{j=1}^{k}a_{j,i}\bigg{)}y_{i}.$

We will omit the index $k$ when the number of variables is clear.

Let us define what we call matrix of higher order, and some basic operations
on it. Hyperdeterminants were first discovered by Arthur Cayley in 1843 (see
[Cay49]).

Let $A$ be any commutative ring. A _matrix of order $k$ and size
$(n_{1},\dots,n_{k})$ on $A$_, where $k$ is a positive integer, and where the
_size_ is a $k$-tuple of positive integers, is a family of elements of $A$
indexed by $[n_{1}]\times\dots\times[n_{k}]$. We can naturally sum two
matrices of the same size by summing corresponding entries.

Let
$C=(c_{i_{1},\dots,i_{k}})_{(i_{1},\dots,i_{k})\in[n_{1}]\times\dots\times[n_{k}]}$
be such a matrix. Let $l\in[k]$ and $P=(p_{i,j})\in\mathcal{M}_{n_{l},m}(A)$
be a usual matrix. Then the _(right) multiplication of $C$ by $P$ along the
$l$-th direction_, denoted by $C\cdot_{l}P$, is the $k$-dimensional matrix $B$
of size

$\operatorname{size}(B)=(n_{1},\dots,n_{l-1},m,n_{l+1},\dots,n_{k}),$

verifying,

$b_{i_{1},\dots,i_{k}}=\sum_{a=1}^{n_{l}}c_{i_{1},\dots,a,\dots,i_{k}}\cdot
p_{a,i_{l}},$

where $a$ is the $l$-th index of $c$.

Let $\operatorname{\mathfrak{C}}_{n}^{k}(A)$ denote the set of _hypercubic
matrices of order $k$ and of size $n$_, i.e., of matrices having size
$(\underbrace{n,\dots,n}_{\text{$k$ times}})$. Let
$C\in\operatorname{\mathfrak{C}}_{n}^{k}(A)$. We define the _hyperdeterminant
of $C$_ by

$\operatorname{det}(C):=\frac{1}{n!}\sum_{\tau_{1},\dots,\tau_{k}\in\mathfrak{S}_{n}}\prod_{j=1}^{k}\sigma(\tau_{j})\prod_{a=1}^{n}c_{\tau_{1}(a),\dots,\tau_{k}(a)},$

where $\mathfrak{S}_{n}$ is the set of permutations on $[n]$, and
$\sigma(\tau)$ denotes the signature of the permutation $\tau$. Notice that
this definition coincides with the usual one for $k=2$. Moreover, the
determinant would be $0$ if $k$ was odd. Let $D$ be the diagonal hypermatrix
$D:=\operatorname{diag}^{k}(a_{1},\dots,a_{n})$ defined by

$D_{i_{1},\dots,i_{k}}=\begin{cases}a_{i_{1}}&\text{if
$i_{1}=\dots=i_{k}$},\\\ 0&\text{otherwise}.\end{cases}$

Then $\operatorname{det}(D)=a_{1}\cdots a_{n}$. The determinant is
multiplicative: if $l\in[k]$ and if $P\in\mathcal{M}_{n}(A)$, then

$\operatorname{det}(C\cdot_{l}P)=\operatorname{det}(C)\operatorname{det}(P).$

Let $u^{1},\dots,u^{k}$ be $k$ families of vectors in $\mathbb{R}^{p}$ of
respective size $l_{1},\dots,l_{k}$. If $(\cdot,\dots,\cdot)_{*}$ is any
$k$-multilinear symmetric product from $(\mathbb{R}^{p})^{k}$ to a commutative
ring, we can extend it to $\operatorname{\bigwedge}\mathbb{R}^{p}$ by

$(\mathbf{u}^{1},\dots,\mathbf{u}^{k})_{*}=\begin{cases}\operatorname{det}((u^{1}_{i_{1}},\dots,u^{k}_{i_{k}})_{*})_{1\leqslant
i_{1},\dots,i_{k}\leqslant l_{1}}&\text{if $l_{1}=\dots=l_{k}$,}\\\
0&\text{otherwise.}\end{cases}$

We allow ourselves to use standard terminologies and notations of inner
products in this case. For instance, $\lVert\mathbf{u}\rVert_{*}^{k}$ denoted
$(\mathbf{u},\dots,\mathbf{u})_{*}$.

For instance, if $l_{1}=\dots=l_{k}=n$, then

$(\mathbf{u}^{1},\mathbf{u}^{2})_{x}=\operatorname{det}((U^{1})^{{}^{\intercal}}X(U^{2})),$

where $X=\operatorname{diag}(x_{1},\dots,x_{n})$, and,

$(\mathbf{u}^{1},\dots,\mathbf{u}^{k})_{x}=\operatorname{det}(X\cdot_{1}U^{1}\cdots\cdot_{k}U^{k}),$

where $X$ is the diagonal hypercubic matrix
$\operatorname{diag}^{k}(x_{1},\dots,x_{n})$ (and $U^{i}$ is the matrix
associated to $u^{i}$).

Notice that $\lVert\cdot\rVert_{x}^{k}$ systematically denotes
$\lVert\cdot\rVert_{2,x}^{k}$.

We state a simple but very useful lemma.

######  Lemma 2.1.

Let $r,p,n$ be three positive integers and $U\in\mathcal{M}_{p,n}(\mathbb{Z})$
be a matrix of rank $r$. Let $F\in\mathcal{M}_{p,r}(\mathbb{Z})$. Assume that
$\mathbb{Z}u\subseteq\mathbb{Z}f$. Then, there exists a unique
$V\in\mathcal{M}_{r,n}(\mathbb{Z})$ such that $U=FV$. Moreover, we have the
following points.

  1. (1)

The following properties are equivalent.

     * •

$f$ is a basis of $\mathbb{Z}{u}$.

     * •

$v^{{}^{\intercal}}$ is a basis of $\langle
u^{{}^{\intercal}}\rangle_{\mathbb{Z}}$.

Futhermore, if they hold, $v$ generates $\mathbb{Z}^{r}$ and
$\lVert\mathbf{u}_{I}\rVert=\lVert\mathbb{Z}u\rVert\cdot\lVert\mathbf{v}_{I}\rVert$
for any subset $I$ of $[n]$ of size $r$.

  2. (2)

The following properties are equivalent.

     * •

$f$ is a basis of $\langle u\rangle_{\mathbb{Z}}$.

     * •

$v^{{}^{\intercal}}$ is a basis of $\mathbb{Z}{u^{{}^{\intercal}}}$.

Futhermore, if they hold, $\lVert\mathbf{u}_{I}\rVert=\lVert\langle
u\rangle_{\mathbb{Z}}\rVert\cdot\lVert\mathbf{v}_{I}\rVert$ for any subset $I$
of $[n]$ of size $r$.

###### Proof.

The first part of the lemma is clear. Notice that columns of $F$, resp. rows
of $V$, have to be free. Let us prove Point (1).

If $f$ is a basis of $\mathbb{Z}{u}$, then there exists
$V^{\prime}\in\mathcal{M}_{n,r}(\mathbb{Z})$ such that
$F=UV^{\prime}=FVV^{\prime}$. Since columns of $F$ are free,
$VV^{\prime}=\operatorname{Id}_{r}$. Thus, $v$ generates $\mathbb{Z}^{r}$. Let
$w$ be such that $w^{{}^{\intercal}}$ is a basis of $\langle
u^{{}^{\intercal}}\rangle_{\mathbb{Z}}$ (which equals $\langle
v^{{}^{\intercal}}\rangle_{\mathbb{Z}}$). This last space contains
$v^{{}^{\intercal}}$. Therefore, there exists a square matrix $G$ such that
$V=GW$. Then,

$1=\operatorname{det}(VV^{\prime})=\operatorname{det}(GWV^{\prime})=\operatorname{det}(G)\operatorname{det}(WV^{\prime}),$

which implies that $G$ is invertible, and then that $v^{{}^{\intercal}}$ is in
fact a basis of $\langle v^{{}^{\intercal}}\rangle_{\mathbb{Z}}$.

Let us prove the converse. If $v^{{}^{\intercal}}$ is a basis of $\langle
u^{{}^{\intercal}}\rangle_{\mathbb{Z}}$, then $v^{{}^{\intercal}}$ can be
completed into a basis ${\widetilde{v}}^{{}^{\intercal}}$ of $\mathbb{Z}^{n}$.
Set $V^{\prime}=\big{(}{\widetilde{V}}^{-1}\big{)}_{[r]}$. As
$UV^{\prime}=FVV^{\prime}=F$, $\mathbb{Z}{f}\subseteq\mathbb{Z}{u}$. The other
inclusion follows from the existence of $V$. Thus, $f$ is in fact a basis of
$\mathbb{Z}u$.

Moreover,

$\lVert\mathbf{u}_{I}\rVert^{2}=\operatorname{det}(U_{I}^{{}^{\intercal}}U_{I})=\operatorname{det}(V_{I}^{{}^{\intercal}}F^{{}^{\intercal}}FV_{I}).$

But $V_{I}$ and $F^{{}^{\intercal}}F$ are square matrices. Thus, since $f$ is
a basis of $\mathbb{Z}u$,

$\operatorname{det}(V_{I}^{{}^{\intercal}}F^{{}^{\intercal}}FV_{I})=\operatorname{det}(V_{I})^{2}\operatorname{det}(F^{{}^{\intercal}}F)=\lVert\mathbf{v}_{I}\rVert^{2}\lVert\mathbb{Z}u\rVert^{2}.$

The second point can be proven in a similar way. ∎

Some specific notations will be introduced later but we can already deal with
the heart of the subject.

###  2.2. Kirchhoff and Symanzik polynomials

In this article, Kirchhoff polynomials are a generalization of polynomials
appearing in the weighted Kirchhoff’s matrix tree theorem, whereas Symanzik
polynomials generalize first and second Symanzik polynomials better known in
Physics (see Examples 2.8, 2.11 and 2.20, Theorem 3.7 and the introduction for
more details). They are dual in a very precise way as we will show at the end
of this subsection, cf. Theorem 2.12.

We will fix some objects for the rest of the section. Let $p,n$ be two
positive integers, $u$ be a family of $n$ elements in $\mathbb{Z}^{p}$, and
$r:=\operatorname{rk}(u)$ be the rank of this family. Recall that $k$ is an
even nonnegative integer.

######  Definition 2.2.

The _Kirchhoff polynomial of order $k$ of $u$_ is defined by

$\operatorname{Kir}_{k}(u;x):=\frac{1}{{\lVert\langle
u\rangle_{\mathbb{Z}}\rVert}^{k}}\sum_{\begin{subarray}{c}I\subseteq[n]\\\
|I|=r\end{subarray}}\lVert\mathbf{u}_{I}\rVert^{k}x^{I}.\qed$

######  Remark 2.3.

Another definition would be obtained replacing $\lVert\langle
u\rangle_{\mathbb{Z}}\rVert$ by $\lVert\mathbb{Z}{u}\rVert$. A priori, no
definition is better than the other. We choose our definition in order to get
determinantal formula without any factor (see Propositions 2.5 and 2.6). As a
counterpart, factors appear in the duality theorem 2.12. ∎

Let $I$ be a subset of $[n]$ of size $r$, let $v$ be such that
$v^{{}^{\intercal}}$ is a basis of $\mathbb{Z}{u^{{}^{\intercal}}}$, and let
$F\in\mathcal{M}_{p,r}(\mathbb{Z})$ be the unique matrix such that $U=FV$. By
Lemma 2.1, $f$ is a basis of $\langle u\rangle_{\mathbb{Z}}$. If
$\mathbf{u}_{I}$ is nonzero, then $\mathbb{Z}{u_{I}}$ is a submodule of
$\langle u\rangle_{\mathbb{Z}}$ of maximal rank, and we therefore have all the
following equalities:

(5) $\frac{\lVert\mathbf{u}_{I}\rVert}{\lVert\langle
u\rangle_{\mathbb{Z}}\rVert}=\lVert\mathbf{v}_{I}\rVert=\lvert\operatorname{det}(V_{I})\rvert=|\langle
u\rangle_{\mathbb{Z}}\big{/}\mathbb{Z}{u_{I}}|=\lvert(\mathbf{v}^{{}^{\intercal}},\mathbf{e}_{I})\rvert,$

where $e$ is the standard basis of $\mathbb{Z}^{n}$. In particular, the
coefficients of $\operatorname{Kir}_{k}$ are in $\mathbb{Z}$.

Moreover, $v^{{}^{\intercal}}$ could be any basis of
$\mathbb{Z}{u^{{}^{\intercal}}}$. But Equation (5) gives a formula of the
coefficients depending on $v$ alone. Thus,

(6) $\operatorname{Kir}_{k}(u;x)$ only depends on
$\mathbb{Z}{u^{{}^{\intercal}}}$.

######  Remark 2.4.

Using $\operatorname{det}(V_{I})$ instead of
$\lVert\mathbf{u}_{I}\rVert/\lVert\langle u\rangle_{\mathbb{Z}}\rVert$ in
Definition 2.2, it is possible to define Kirchhoff polynomials for $k$ odd, or
even for $u$ a family of vectors over $A^{n}$ where $A$ is any PID. However,
one has to be careful because the $k$-th power of an element of $A^{*}$ could
be different from $1$. The polynomial depends on the chosen $v$ (up to a
$k$-th power of an invertible of $A^{*}$). Moreover, one has to add some
artificial signs to the Symanzik polynomial (see Definition 2.9) in order to
make Theorem 2.12 about the duality true. For more details, we refer to
[Piq17]. ∎

Kirchhoff polynomials of order $2$ have a more computable definition under the
condition $r=p$.

######  Proposition 2.5.

If $r=p$ (or equivalently if $u^{{}^{\intercal}}$ is free), then

$\operatorname{Kir}_{2}(u;x)=\lVert\mathbf{u}^{{}^{\intercal}}\rVert_{x}^{2}=\operatorname{det}(UXU^{{}^{\intercal}}),$

where $X\in\mathcal{M}_{n}(\mathbb{Z}[x])$ is the diagonal matrix
$\operatorname{diag}(x_{1},\dots,x_{n})$.

###### Proof.

The second equality comes directly from the definitions. Let $e$ be the
standard basis of $\mathbb{R}^{n}$. As the standard orthonormal basis of
$\operatorname{\bigwedge}^{r}\mathbb{R}^{n}$ (for the standard inner product)
is an orthogonal basis for $(\cdot,\cdot)_{x}$, we obtain,

$\displaystyle\lVert\mathbf{u}^{{}^{\intercal}}\rVert_{x}^{2}$
$\displaystyle=\sum_{\begin{subarray}{c}I\subseteq[n]\\\
|I|=r\end{subarray}}(\mathbf{u}^{{}^{\intercal}},\mathbf{e}_{I})^{2}\lVert\mathbf{e}_{I}\rVert_{x}^{2}$
$\displaystyle=\sum_{\begin{subarray}{c}I\subseteq[n]\\\
|I|=r\end{subarray}}\lVert\mathbf{u}_{I}\rVert^{2}x^{I}.$

It remains to check that $\lVert\langle u\rangle_{\mathbb{Z}}\rVert=1$, but
this is true because $\langle u\rangle_{\mathbb{Z}}=\mathbb{Z}^{p}$. ∎

For orders larger than 2, one can obtain a similar formula.

######  Proposition 2.6.

If $r=p$ (or equivalently if $u^{{}^{\intercal}}$ is free), then

$\operatorname{Kir}_{k}(u;x)=\lVert\mathbf{u}^{{}^{\intercal}}\rVert_{k,x}^{k}=\operatorname{det}\big{(}X\cdot_{1}U^{{}^{\intercal}}\cdot_{2}\cdots\cdot_{k}U^{{}^{\intercal}}\big{)},$

where $X$ is the diagonal hypercubic matrix
$\operatorname{diag}^{k}(x_{1},\dots,x_{n})$.

###### Proof.

The same proof works if one replaces the computation by

$\displaystyle\lVert\mathbf{u}^{{}^{\intercal}}\rVert_{k,x}^{k}$
$\displaystyle=\sum_{\begin{subarray}{c}I\subseteq[n]\\\
|I|=r\end{subarray}}(\mathbf{u}^{{}^{\intercal}},\mathbf{e}_{I})^{k}\lVert\mathbf{e}_{I}\rVert_{k,x}^{k}$
$\displaystyle=\sum_{\begin{subarray}{c}I\subseteq[n]\\\
|I|=r\end{subarray}}\lVert\mathbf{u}_{I}\rVert^{k}x^{I}.\qed$

######  Example 2.7.

How to compute the Kirchhoff polynomial of any family $u$ with the
determinantal formula? Take $v$ a family such that $v^{{}^{\intercal}}$ is a
basis of $\mathbb{Z}{u^{{}^{\intercal}}}$. Let $F$ be the unique matrix such
that $U=FV$ (it is not necessary to compute $f$). By Lemma 2.1, for any subset
$I\subseteq[n]$ of size $r$,

$\lVert\mathbf{u}_{I}\rVert=\lVert\mathbf{v}_{I}\rVert\lVert\langle
u\rangle_{\mathbb{Z}}\rVert.$

As $\langle v\rangle_{\mathbb{Z}}=\mathbb{Z}^{r}$, one can rewrite last
equation as

$\frac{\lVert\mathbf{u}_{I}\rVert}{\lVert\langle
u\rangle_{\mathbb{Z}}\rVert}=\frac{\lVert\mathbf{v}_{I}\rVert}{\lVert\langle
v\rangle_{\mathbb{Z}}\rVert}.$

Thus,

$\operatorname{Kir}_{k}(u;x)=\operatorname{Kir}_{k}(v;x)=\lVert\mathbf{v}^{{}^{\intercal}}\rVert_{k,x}^{k}.$

Note that, if instead of taking a basis of $\mathbb{Z}{u^{{}^{\intercal}}}$,
one takes a basis of $\langle u^{{}^{\intercal}}\rangle_{\mathbb{Z}}$, you
will divide all coefficients by a common divisor equal to

$\bigg{(}\frac{\lVert\mathbb{Z}{u}\rVert}{\lVert\langle
u\rangle_{\mathbb{Z}}\rVert}\bigg{)}^{k}.$

We will see a meaning of this factor in Remark 3.8. ∎

######  Example 2.8.

Now let us explain more precisely the link between Kirchhoff as mathematician
and Kirchhoff polynomials. Let $G=(V,E)$ be a connected graph with vertex set
$V$ of size $p$ and edge set $E$ of size $n$. Suppose that vertices are
labelled from $1$ to $p$ and edges from $1$ to $n$. By abuse of notation, the
same letter will denote a vertex, resp. an edge, and its label. Similarly, a
set of edges could denote a set of numbers. Let
$Q=(q_{v,e})\in\mathcal{M}_{p,n}(\mathbb{Z})$ be an incidence matrix of $G$,
i.e., put an orientation on edges of $G$ and set, for each $v\in V$ and each
$e\in E$,

$q_{v,e}:=\left\\{\begin{array}[]{ll}0&\text{if $e$ is a loop,}\\\ 1&\text{if
$v$ is the head of $e$,}\\\ -1&\text{if $v$ is the tail of $e$,}\\\ 0&\text{if
$v$ and $e$ are not incident.}\end{array}\right.$

Let $i\in[p]$ be any number, and $U$ be the matrix $Q$ where we deleted the
$i$-th row. The well-known Kirchhoff’s matrix-tree theorem, in its weighted
form, states that

$\operatorname{det}(UXU^{{}^{\intercal}})=\sum_{I\in\mathcal{T}}x^{I},$

where $X=\operatorname{diag}(x_{1},\dots,x_{n})$, and
$\mathcal{T}\subseteq\operatorname{\mathcal{P}}(E)$ is the family of all
subsets $I\subseteq E$ that verify that the spanning subgraph of $G$ with edge
set $I$ is a spanning tree of $G$. Then Proposition 2.5 implies that

$\operatorname{Kir}_{2}(u;x)=\sum_{I\in\mathcal{T}}x^{I}.$

In fact, in this very special case, $\operatorname{Kir}_{k}(u;x)$ does not
depend on $k$. Moreover one can see that
$\mathbb{Z}{q^{{}^{\intercal}}}=\mathbb{Z}{u^{{}^{\intercal}}}$. Thus, by (6),
we even have

(7) $\operatorname{Kir}_{k}(q;x)=\sum_{I\in\mathcal{T}}x^{I}.$

We will see later, in Theorem 3.7, a generalization of Kirchhoff’s theorem to
the case of finite simplicial complexes. ∎

Symanzik polynomials have a similar definition, but note the complement in the
last exponent.

######  Definition 2.9.

The _Symanzik polynomial of order $k$ of $u$_ is defined by

$\operatorname{Sym}_{k}(u;x):=\frac{1}{{\lVert\langle
u\rangle_{\mathbb{Z}}\rVert}^{k}}\sum_{\begin{subarray}{c}I\subseteq[n]\\\
|I|=r\end{subarray}}\lVert\mathbf{u}_{I}\rVert^{k}x^{I^{c}}.\qed$

######  Remark 2.10.

Kirchhoff and Symanzik polynomials are so similar that one can easily define
ones from the others thanks to following formulæ.

$\displaystyle\operatorname{Sym}_{k}(u;x)=x_{1}\cdots
x_{n}\operatorname{Kir}_{k}(u;(x_{1}^{-1},\dots,x_{n}^{-1})),$
$\displaystyle\operatorname{Kir}_{k}(u;x)=x_{1}\cdots
x_{n}\operatorname{Sym}_{k}(u;(x_{1}^{-1},\dots,x_{n}^{-1})).\qed$

######  Example 2.11.

What is the link between Symanzik polynomials of Definition 2.9 and the first
Symanzik polynomial of the introduction (1)? Let $G$ be a graph as defined in
Example 2.8. Set $k=2$. We have seen in Example 2.8 that, using the same
notations,

$\operatorname{Kir}_{k}(q;x)=\sum_{I\in\mathcal{T}}x^{I}.$

Using Remark 2.10, we obtain

$\displaystyle\operatorname{Sym}_{k}(q;x)$ $\displaystyle=x_{1}\cdots
x_{n}\operatorname{Kir}_{k}(q;(x_{1}^{-1},\dots,x_{n}^{-1}))$
$\displaystyle=\sum_{I\in\mathcal{T}}x^{I^{c}},$

which is exactly the first Symanzik polynomial. ∎

Now we can state the duality theorem.

######  Theorem 2.12 (Duality).

Let $q$ be a positive integer and $v$ be a family of $n$ vectors in
$\mathbb{Z}^{q}$ such $v^{{}^{\intercal}}$ spans $\ker(U)$. Then

(8)
$\frac{1}{a^{k}}\operatorname{Sym}_{k}(u;x)=\frac{1}{b^{k}}\operatorname{Kir}_{k}(v;x),$

where

$\displaystyle a$
$\displaystyle:=\frac{\lVert\mathbb{Z}{u}\rVert}{\lVert\langle
u\rangle_{\mathbb{Z}}\rVert}=|\langle
u\rangle_{\mathbb{Z}}\big{/}\mathbb{Z}{u}|\text{ and}$ $\displaystyle b$
$\displaystyle:=\frac{\lVert\mathbb{Z}{v}\rVert}{\lVert\langle
v\rangle_{\mathbb{Z}}\rVert}=|\langle
v\rangle_{\mathbb{Z}}\big{/}\mathbb{Z}{v}|.$

Note that $\ker(U)$ denotes a vector subspace of $\mathbb{R}^{n}$, and that
$\ker(U)=\langle u^{{}^{\intercal}}\rangle^{\perp}$.

###### Proof.

Set $\widetilde{u}$ a family of vectors such that
${\widetilde{u}}^{{}^{\intercal}}$ is a basis of $\langle
u^{{}^{\intercal}}\rangle_{\mathbb{Z}}$. One can find a matrix
$U^{\prime}\in\mathcal{M}_{r,n}(\mathbb{Z})$ such that
$\widetilde{U}U^{\prime{{}^{\intercal}}}=\operatorname{Id}_{r}$. Similarly,
set $\widetilde{v}$ such that ${\widetilde{v}}^{{}^{\intercal}}$ is a basis of
$\langle v^{{}^{\intercal}}\rangle_{\mathbb{Z}}$, and
$V^{\prime}\in\mathcal{M}_{n-r,n}(\mathbb{Z})$ such that
$\widetilde{V}V^{\prime{{}^{\intercal}}}=\operatorname{Id}_{n-r}$.

Let $I$ be a subset of $[n]$ of size $r$. The coefficient of $x^{I^{c}}$ in
the left-hand side of (8) is

$\Big{(}\frac{\lVert\langle
u\rangle_{\mathbb{Z}}\rVert}{\lVert\mathbb{Z}{u}\rVert}\frac{\lVert\mathbf{u}_{I}\rVert}{\lVert\langle
u\rangle_{\mathbb{Z}}\rVert}\Big{)}^{k}.$

Therefore, by Lemma 2.1, this coefficient is equal to
$\lVert\mathbf{\widetilde{u}}_{I}\rVert^{k}$. Applying the same argument on
the right-hand side, it remains to prove that

$\lVert\mathbf{\widetilde{u}}_{I}\rVert=\lVert\mathbf{\widetilde{v}}_{I^{c}}\rVert.$

We will deduce this last equality from

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}\color[rgb]{0.75,0.75,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0.75,0.75}\pgfsys@color@gray@stroke{0.75}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0.75}\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{0.75,0.75,0.75}{}\pgfsys@moveto{21.90874pt}{0.0pt}\pgfsys@moveto{21.90874pt}{0.0pt}\pgfsys@lineto{21.90874pt}{19.91693pt}\pgfsys@lineto{37.84233pt}{19.91693pt}\pgfsys@lineto{37.84233pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{37.84233pt}{19.91693pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} {\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{27.37553pt}{6.73625pt}\pgfsys@invoke{
}\hbox{{\definecolor[named]{.}{rgb}{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\hbox{{\definecolor[named]{.}{rgb}{0,0,0}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}$0$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{{}{}}{{}}{}{}{}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0.75,0.75,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0.75,0.75}\pgfsys@color@gray@stroke{0.75}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0.75}\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{0.75,0.75,0.75}{}\pgfsys@moveto{0.0pt}{-1.9918pt}\pgfsys@moveto{0.0pt}{-1.9918pt}\pgfsys@lineto{0.0pt}{-17.9254pt}\pgfsys@lineto{19.91693pt}{-17.9254pt}\pgfsys@lineto{19.91693pt}{-1.9918pt}\pgfsys@closepath\pgfsys@moveto{19.91693pt}{-17.9254pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} {\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{7.45847pt}{-12.28499pt}\pgfsys@invoke{
}\hbox{{\definecolor[named]{.}{rgb}{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\hbox{{\definecolor[named]{.}{rgb}{0,0,0}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}$*$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{{}{}}{{}}{}{}{}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0.75,0.75,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0.75,0.75}\pgfsys@color@gray@stroke{0.75}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0.75}\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{0.75,0.75,0.75}{}\pgfsys@moveto{21.90874pt}{-1.9918pt}\pgfsys@moveto{21.90874pt}{-1.9918pt}\pgfsys@lineto{21.90874pt}{-17.9254pt}\pgfsys@lineto{37.84233pt}{-17.9254pt}\pgfsys@lineto{37.84233pt}{-1.9918pt}\pgfsys@closepath\pgfsys@moveto{37.84233pt}{-17.9254pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} {\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{23.97969pt}{-12.82805pt}\pgfsys@invoke{
}\hbox{{\definecolor[named]{.}{rgb}{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\hbox{{\definecolor[named]{.}{rgb}{0,0,0}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}$\operatorname{Id}_{s}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\right),$

where $s:=n-r$, and from

$\left(\,\raisebox{-0.45pt}{ \leavevmode\hbox to38.33pt{\vbox
to38.24pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-18.1254pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{}{ {}{{}}{}
{{}{}}{{}}{}{}{}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0.75,0.75,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0.75,0.75}\pgfsys@color@gray@stroke{0.75}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0.75}\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{0.75,0.75,0.75}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{19.91693pt}\pgfsys@lineto{19.91693pt}{19.91693pt}\pgfsys@lineto{19.91693pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{19.91693pt}{19.91693pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} {\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{5.22946pt}{6.7207pt}\pgfsys@invoke{
}\hbox{{\definecolor[named]{.}{rgb}{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\hbox{{\definecolor[named]{.}{rgb}{0,0,0}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}$U^{\prime}_{I}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{{}{}}{{}}{}{}{}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0.75,0.75,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0.75,0.75}\pgfsys@color@gray@stroke{0.75}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0.75}\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{0.75,0.75,0.75}{}\pgfsys@moveto{21.90874pt}{0.0pt}\pgfsys@moveto{21.90874pt}{0.0pt}\pgfsys@lineto{21.90874pt}{19.91693pt}\pgfsys@lineto{37.84233pt}{19.91693pt}\pgfsys@lineto{37.84233pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{37.84233pt}{19.91693pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} {\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{25.14653pt}{6.89293pt}\pgfsys@invoke{
}\hbox{{\definecolor[named]{.}{rgb}{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\hbox{{\definecolor[named]{.}{rgb}{0,0,0}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}$U^{\prime}_{I^{c}}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{{}{}}{{}}{}{}{}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0.75,0.75,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0.75,0.75}\pgfsys@color@gray@stroke{0.75}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0.75}\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{0.75,0.75,0.75}{}\pgfsys@moveto{0.0pt}{-1.9918pt}\pgfsys@moveto{0.0pt}{-1.9918pt}\pgfsys@lineto{0.0pt}{-17.9254pt}\pgfsys@lineto{19.91693pt}{-17.9254pt}\pgfsys@lineto{19.91693pt}{-1.9918pt}\pgfsys@closepath\pgfsys@moveto{19.91693pt}{-17.9254pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} {\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{5.73013pt}{-12.61305pt}\pgfsys@invoke{
}\hbox{{\definecolor[named]{.}{rgb}{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\hbox{{\definecolor[named]{.}{rgb}{0,0,0}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}$\widetilde{V}_{I}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{{}{}}{{}}{}{}{}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0.75,0.75,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0.75,0.75}\pgfsys@color@gray@stroke{0.75}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0.75}\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{0.75,0.75,0.75}{}\pgfsys@moveto{21.90874pt}{-1.9918pt}\pgfsys@moveto{21.90874pt}{-1.9918pt}\pgfsys@lineto{21.90874pt}{-17.9254pt}\pgfsys@lineto{37.84233pt}{-17.9254pt}\pgfsys@lineto{37.84233pt}{-1.9918pt}\pgfsys@closepath\pgfsys@moveto{37.84233pt}{-17.9254pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} {\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{24.95479pt}{-12.44083pt}\pgfsys@invoke{
}\hbox{{\definecolor[named]{.}{rgb}{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\hbox{{\definecolor[named]{.}{rgb}{0,0,0}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}$\widetilde{V}_{I^{c}}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\right)\left(\,\raisebox{-0.45pt}{
\leavevmode\hbox to39.3pt{\vbox to39.72pt{\pgfpicture\makeatletter\hbox{\hskip
0.2pt\lower-19.60384pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{}{ {}{{}}{}
{{}{}}{{}}{}{}{}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0.75,0.75,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0.75,0.75}\pgfsys@color@gray@stroke{0.75}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0.75}\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{0.75,0.75,0.75}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{19.91693pt}\pgfsys@lineto{19.91693pt}{19.91693pt}\pgfsys@lineto{19.91693pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{19.91693pt}{19.91693pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} {\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{5.73013pt}{5.7318pt}\pgfsys@invoke{
}\hbox{{\definecolor[named]{.}{rgb}{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\hbox{{\definecolor[named]{.}{rgb}{0,0,0}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}$\widetilde{U}_{I}^{{}^{\intercal}}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{{}{}}{{}}{}{}{}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0.75,0.75,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0.75,0.75}\pgfsys@color@gray@stroke{0.75}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0.75}\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{0.75,0.75,0.75}{}\pgfsys@moveto{21.90874pt}{0.0pt}\pgfsys@moveto{21.90874pt}{0.0pt}\pgfsys@lineto{21.90874pt}{19.91693pt}\pgfsys@lineto{37.84233pt}{19.91693pt}\pgfsys@lineto{37.84233pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{37.84233pt}{19.91693pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} {\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{27.37553pt}{6.73625pt}\pgfsys@invoke{
}\hbox{{\definecolor[named]{.}{rgb}{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\hbox{{\definecolor[named]{.}{rgb}{0,0,0}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}$0$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
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Indeed, this implies

$\displaystyle\lVert\mathbf{\widetilde{u}}^{{}^{\intercal}}\wedge\mathbf{v^{\prime}}^{{}^{\intercal}}\rVert\cdot\lVert\mathbf{\widetilde{v}}_{I^{c}}\rVert=\lVert\mathbf{\widetilde{u}}_{I}\rVert,$
$\displaystyle\lVert\mathbf{u^{\prime}}^{{}^{\intercal}}\wedge\mathbf{\widetilde{v}}^{{}^{\intercal}}\rVert\cdot\lVert\mathbf{\widetilde{u}}_{I}\rVert=\lVert\mathbf{\widetilde{v}}_{I^{c}}\rVert.$

Since all these factors are nonnegative integers. We must have

$\lVert\mathbf{\widetilde{u}}_{I}\rVert=\lVert\mathbf{\widetilde{v}}_{I^{c}}\rVert,$

which concludes the proof. ∎

######  Remark 2.13.

In the proof above, the equality of coefficients implies that

$\frac{\lVert\mathbf{u}_{I}\rVert}{\lVert\mathbb{Z}{u}\rVert}=\frac{\lVert\mathbf{v}_{I^{c}}\rVert}{\lVert\mathbb{Z}{v}\rVert}.$

In fact, one can prove something stronger, namely,

$\mathbb{Z}{u}\big{/}\mathbb{Z}{u_{I}}\simeq\mathbb{Z}{v}\big{/}\mathbb{Z}{v_{I^{c}}}.\qed$

We need a lemma for the proof of Theorem 5.1. We state this lemma here because
its proof is very similar to that of Theorem 2.12. If $I$ is a subset of $[n]$
and if $\underline{I}$ is an ordering of the elements of $I$, then we write
$\varepsilon(\underline{I}):=(\mathbf{e}_{\underline{I}},\mathbf{e}_{I})$,
i.e., $(-1)^{\iota}$ where $\iota$ is the number of inversions in
$\underline{I}$.

######  Lemma 2.14.

Let $p,m$ be two positive integers. Let $\widetilde{u}$ and $\widetilde{v}$ be
such that $\widetilde{u}^{{}^{\intercal}}$ and
$\widetilde{v}^{{}^{\intercal}}$ are two free families of vectors in
$\mathbb{R}^{m}$ of respective size $p$ and $m-p$. Assume that
$\widetilde{U}{\widetilde{V}}^{{}^{\intercal}}=0$. Let $I$ be a subset of
$[m]$ of size $p$. Then

$\operatorname{det}(\widetilde{U}_{I})=\lambda\,\varepsilon(I\star
I^{c})\operatorname{det}(\widetilde{V}_{I^{c}}),$

where $\lambda\in\mathbb{R}^{*}$ does not depend on $I$.

###### Proof.

We define $V^{\prime}\in\mathcal{M}_{m-p,m}(\mathbb{R})$ as in the proof of
Theorem 2.12, i.e., such that
$\widetilde{V}{V^{\prime}}^{{}^{\intercal}}=\operatorname{Id}_{m-p}$. We still
get Equation (9). By reordering columns and by transposing, this equation
leads to

$\varepsilon(I\star
I^{c})\operatorname{det}(\widetilde{U}^{{}^{\intercal}}\star
V^{\prime{{}^{\intercal}}})\operatorname{det}(\widetilde{V}_{I^{c}})=\operatorname{det}(\widetilde{U}_{I}).$

Then set $\lambda=\operatorname{det}(\widetilde{U}^{{}^{\intercal}}\star
V^{\prime{{}^{\intercal}}})$ to conclude the proof ($\lambda$ cannot be $0$
since there exists an $I$ such that $\operatorname{det}(\widetilde{U}_{I})\neq
0$). ∎

Now let us extend determinantal formulæ to Symanzik polynomials. From the
determinantal formula for Kirchhoff polynomials (Propositions 2.5 and 2.6) and
from the duality theorem (Theorem 2.12), we immediately obtain the following
formula.

######  Proposition 2.15.

If $v$ is a family of vectors in $\mathbb{Z}^{n-r}$ such that
$v^{{}^{\intercal}}$ is a basis of the vector subspace $\ker(U)$, then

$\operatorname{Sym}_{k}(u;x)=\Big{(}\frac{a}{b}\Big{)}^{k}\lVert\mathbf{v}^{{}^{\intercal}}\rVert_{k,x}^{k}$

where $a=\lVert\mathbb{Z}u\rVert/\lVert\langle u\rangle_{\mathbb{Z}}\rVert$
and $b=\lVert\mathbb{Z}v\rVert/\lVert\langle v\rangle_{\mathbb{Z}}\rVert$ are
defined as in Theorem 2.12.

###  2.3. Symanzik polynomials with parameters

Now we want to generalize the second Symanzik polynomials defined in the
introduction (2). Actually, one can naturally add more than one parameter.

######  Definition 2.16.

Let $l$ be a nonnegative integer and let $w=(w_{1},\dots,w_{l})$ be a family
of elements in $\langle u\rangle$. The _Symanzik polynomial of order $k$ of
$u$ with parameters $w_{1},\dots,w_{l}$_ is defined by

$\operatorname{Sym}_{k}(u;w;x):=\frac{1}{{\lVert\langle
u\rangle_{\mathbb{Z}}\rVert}^{k}}\sum_{\begin{subarray}{c}I\subseteq[n]\\\
|I|=r-l\end{subarray}}\lVert\mathbf{u}_{I}\wedge\mathbf{w}\rVert^{k}x^{I^{c}}.\qed$

######  Remark 2.17.

In the case $w_{1},\dots,w_{l}\in\mathbb{Z}u$, one simply has

$\operatorname{Sym}_{k}(u;w;x)=\operatorname{Sym}_{k}(u\star
w;x\star(\underbrace{0,\dots,0}_{\text{$l$ times}})).\qed$

Once more, there is a determinantal formula.

######  Proposition 2.18.

Let $l$ be a nonnegative integer and $w$ be a family of $l$ elements in
$\langle u\rangle$. For $i\in[l]$, let $\widetilde{w}_{i}\in\mathbb{R}^{n}$ be
such that $w_{i}=U\widetilde{w}_{i}$. Let $v$ be a family of vectors in
$\mathbb{Z}^{n-r}$ such that $v^{{}^{\intercal}}$ is a basis of the vector
subspace $\ker(U)$. Then,

$\operatorname{Sym}_{k}(u;w;x)=\Big{(}\frac{a}{b}\Big{)}^{k}\lVert\mathbf{v}^{{}^{\intercal}}\wedge\mathbf{\widetilde{w}}\rVert_{k,x}^{k}$

where $a=\lVert\mathbb{Z}u\rVert/\lVert\langle u\rangle_{\mathbb{Z}}\rVert$
and $b=\lVert\mathbb{Z}v\rVert/\lVert\langle v\rangle_{\mathbb{Z}}\rVert$ are
defined as in Theorem 2.12.

###### Proof.

By the multilinearity of the determinant and by a density argument, we can
restrict our study to the case $w_{1},\dots,w_{l}\in\mathbb{Z}u$. Applying
Proposition 2.15 to $\operatorname{Sym}_{k}(u\star w;x\star(0,\dots,0))$, one
obtains that

$\operatorname{Sym}_{k}(u;w;x)=\Big{(}\frac{a^{\prime}}{b^{\prime}}\Big{)}^{k}\lVert\mathbf{v}^{\prime{{}^{\intercal}}}\rVert_{k,x^{\prime}}^{k}\;,$

where

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$\displaystyle x^{\prime}=(x_{1},\dots,x_{n},0,\dots,0),$ $\displaystyle
a^{\prime}=\frac{\lVert\mathbb{Z}{(u\star w)}\rVert}{\lVert\langle u\star
w\rangle_{\mathbb{Z}}\rVert}\text{ and}$ $\displaystyle
b^{\prime}=\frac{\lVert\mathbb{Z}{v^{\prime}}\rVert}{\lVert\langle
v^{\prime}\rangle_{\mathbb{Z}}\rVert}.$

It is not difficult to see that $a^{\prime}=a$, $b^{\prime}=b$ and
$\lVert\mathbf{v}^{\prime{{}^{\intercal}}}\rVert_{k,x^{\prime}}^{k}=\lVert\mathbf{v}^{{}^{\intercal}}\wedge\widetilde{\mathbf{w}}\rVert_{k,x}^{k}$.
∎

We now give another form of Symanzik polynomials which will allow us to
explain the link between Definition 2.16 and the second Symanzik polynomial of
the introduction (2). We need to define the following orientation. Choose a
volume form $\omega$ on $\langle u\rangle$ and, for two subsets $I$ and $J$ of
respective sizes $r-l$ and $l$, set
$\varepsilon_{I}(J):=\operatorname{sign}(\omega(\mathbf{u}_{I\star
J}))\in\\{-1,0,1\\}$. Depending on $\omega$, we will get an orientation or its
opposite.

######  Proposition 2.19.

Let $l$ be a nonnegative integer and $w$ be a family of $l$ elements in
$\langle u\rangle$. Then,

$\operatorname{Sym}_{k}(u;w;x)=\frac{1}{{\lVert\langle
u\rangle_{\mathbb{Z}}\rVert}^{k}}\sum_{\begin{subarray}{c}I\subseteq[n]\\\
|I|=r-l\end{subarray}}\Big{(}\sum_{\begin{subarray}{c}J\subseteq I^{c}\\\
|J|=l\end{subarray}}\varepsilon_{I}(J)\lVert\mathbf{u}_{I\cup
J}\rVert(\mathbf{\widetilde{w}},\mathbf{e}_{J})\Big{)}^{k}x^{I^{c}},$

where $e$ is the standard basis of $\mathbb{R}^{n}$ and $\widetilde{w}$ is
defined as in Proposition 2.18.

Note that, since $k$ is even, the right-hand side does not depend on $\omega$.

###### Proof.

Without loss of generality, assume that members of $w$ are in $\mathbb{Z}u$.
Let $f$ be a basis of $\mathbb{Z}u=\mathbb{Z}(u\star w)$ such that
$\omega(\mathbf{f})>0$. Let $V\in\mathcal{M}_{r,n}(\mathbb{Z})$ be such that
$U=FV$. Then,
$\varepsilon_{I}(J)=\operatorname{sign}(\operatorname{det}(V_{I\star J}))$.
Let $T=V\widetilde{W}$. Lemma 2.1 shows that, for $I\subseteq[n]$ of size
$r-l$, we have

(10)
$\lVert\mathbf{u}_{I}\wedge\mathbf{w}\rVert^{k}=\lVert\mathbb{Z}u\rVert^{k}\lVert\mathbf{v}_{I}\wedge\mathbf{t}\rVert^{k}=\lVert\mathbb{Z}u\rVert^{k}(\mathbf{v}^{{}^{\intercal}},\mathbf{e}_{I}\wedge\mathbf{\widetilde{w}})^{k}.$

Then,

$\displaystyle(\mathbf{v}^{{}^{\intercal}},\mathbf{e}_{I}\wedge\mathbf{\widetilde{w}})$
$\displaystyle=\bigg{(}\mathbf{v}^{{}^{\intercal}},\mathbf{e}_{I}\wedge\Big{(}\sum_{\begin{subarray}{c}J\subseteq[n]\\\
|J|=l\end{subarray}}(\mathbf{\widetilde{w}},\mathbf{e}_{J})\mathbf{e}_{J}\Big{)}\bigg{)}$
$\displaystyle=\sum_{\begin{subarray}{c}J\subseteq I^{c}\\\
|J|=l\end{subarray}}(\mathbf{v}^{{}^{\intercal}},\mathbf{e}_{I}\wedge\mathbf{e}_{J})(\mathbf{\widetilde{w}},\mathbf{e}_{J})$
$\displaystyle=\sum_{\begin{subarray}{c}J\subseteq I^{c}\\\
|J|=l\end{subarray}}\varepsilon_{I}(J)\lVert\mathbf{v}_{I\cup
J}\rVert(\mathbf{\widetilde{w}},\mathbf{e}_{J}).$

Hence, by (10),

$
\lVert\mathbf{u}_{I}\wedge\mathbf{w}\rVert^{k}=\lVert\mathbb{Z}u\rVert^{k}\bigg{(}\sum_{\begin{subarray}{c}J\subseteq
I^{c}\\\ |J|=l\end{subarray}}\varepsilon_{I}(J)\lVert\mathbf{v}_{I\cup
J}\rVert(\mathbf{\widetilde{w}},\mathbf{e}_{J})\bigg{)}^{k}.$

By Lemma 2.1, $\lVert\mathbb{Z}u\rVert\lVert\mathbf{v}_{I\cup
J}\rVert=\lVert\mathbf{u}_{I\cup J}\rVert$. We conclude the proof using
Definition 2.16. ∎

######  Example 2.20.

In this example, we will explain the link between the second Symanzik
polynomial from the introduction and Symanzik polynomials with one parameter
introduced above. More precisely, we want to show that

(11) $\phi_{G}(\mathfrak{p},x)=\operatorname{Sym}_{2}(q;(\mathfrak{p});x),$

where we recall notations below.

Let $G=(V,E)$ be a connected graph as in Example 2.8, and
$Q\in\mathcal{M}_{p,n}(\mathbb{Z})$ be an incidence matrix of $G$. Take
$\mathfrak{p}$ a vector in $\mathbb{R}^{p}$ such that $\sum_{v\in
V}\mathfrak{p}_{v}=0$ (this is a slightly less general case compared to the
definition in the introduction since we take $D=1$). The second Symanzik
polynomial, denoted by $\phi_{G}$, is defined by

$\phi_{G}(\mathfrak{p},x):=\sum_{F\in\operatorname{\mathcal{SF}}_{2}}-\mathfrak{p}_{F_{1}}\mathfrak{p}_{F_{2}}\prod_{e\not\in
F}x_{e},$

with the following notations. The set $\operatorname{\mathcal{SF}}_{2}$
denotes the set of spanning forests of $G$ that have two connected components.
For $F\in\operatorname{\mathcal{SF}}_{2}$, $F_{1}$ and $F_{2}$ denotes the two
connected components of $F$. Then, $\mathfrak{p}_{F_{i}}:=\sum_{v\in
F_{i}}\mathfrak{p}_{v}$ for $i\in\\{1,2\\}$. Since the sum of
$\mathfrak{p}_{v}$ is zero, one can replace
$-\mathfrak{p}_{F_{1}}\mathfrak{p}_{F_{2}}$ by $\mathfrak{p}_{F_{1}}^{2}$.

Since $G$ is a connected graph, $\sum_{v\in V}\mathfrak{p}_{v}=0$ is
equivalent to $\mathfrak{p}\in\langle q\rangle$. Now, we compare
$\phi_{G}(\mathfrak{p},x)$ with the Symanzik polynomial
$\operatorname{Sym}_{2}(q;(\mathfrak{p});x)$. By Proposition 2.19, with
similar notations,

$\operatorname{Sym}_{2}(q;(\mathfrak{p});x)=\frac{1}{\lVert\langle
q\rangle_{\mathbb{Z}}\rVert^{2}}\sum_{\begin{subarray}{c}I\subseteq[n]\\\
|I|=r-1\end{subarray}}\Big{(}\sum_{\begin{subarray}{c}j\in
I^{c}\end{subarray}}\varepsilon_{I}(j)\lVert\mathbf{q}_{I+j}\rVert\widetilde{\mathfrak{p}}_{j}\Big{)}^{2}x^{I^{c}},$

where we write $\varepsilon_{I}(j)$ instead of $\varepsilon_{I}((j))$. By (7),
the ratio $\lVert\mathbf{q}_{I+j}\rVert^{2}/\lVert\langle
q\rangle_{\mathbb{Z}}\rVert^{2}$ equals $1$ if $I+j$ forms a spanning tree of
$G$, and $0$ otherwise. Thus, we can restrict the sum to the sets $I$ that
correspond to $2$-forests of $G$. It remains to show that, if $I\subseteq[n]$
corresponds to $F\in\operatorname{\mathcal{SF}}_{2}$, then

$\Big{\lvert}\sum_{\begin{subarray}{c}j\in
I^{c}\end{subarray}}\varepsilon_{I}(j)\widetilde{\mathfrak{p}}_{j}\Big{\rvert}=\lvert\mathfrak{p}_{F_{1}}\rvert.$

Let us try to better understand what happens here. On each edge $e$, we put a
weight $\widetilde{\mathfrak{p}}_{e}$ such that, for each vertex $v$,
$\mathfrak{p}_{v}$ equals the algebraic sum of the weight of the edges
incident to $v$:

$\mathfrak{p}_{v}=\sum_{e\in E}q_{v,e}\widetilde{\mathfrak{p}}_{e}.$

Thus, when we sum $\mathfrak{p}_{v}$ over all vertices $v$ of $F_{1}$, each
edge in $F_{1}$ appears two times with opposite signs and each edge in $F_{2}$
does not appear in the sum. Thus, it only remains edges joining $F_{1}$ and
$F_{2}$:

$\mathfrak{p}_{F_{1}}=\sum_{e\in
E(F_{1},F_{2})}\varepsilon_{F_{2},F_{1}}(e)\widetilde{\mathfrak{p}}_{e},$

where $E(F_{1},F_{2})$ is the set of edges joining $F_{1}$ and $F_{2}$, and
$\varepsilon_{F_{2},F_{1}}(e)$ equals $+1$ if $e$ is oriented from $F_{2}$ to
$F_{1}$, and equals $-1$ otherwise. Now, let us see how $\varepsilon_{I}$
computes this orientation.

More precisely, it remains to show that, if $e$ and $e^{\prime}$ are two edges
of $E(F_{1},F_{2})$, then $e$ and $e^{\prime}$ have the same orientation
(between $F_{1}$ and $F_{2}$) if and only if
$\varepsilon_{I}(e)=\varepsilon_{I}(e^{\prime})$. Suppose that they do not
have the same orientation. In this case, there exists an oriented cycle in
$F+e+e^{\prime}$ respecting their orientation. This means that, for some
nonnegative integer $l$, there exist $i_{1},\dots,i_{l}\in I$ and
$\eta_{1},\dots,\eta_{l}\in\\{-1,+1\\}$ such that

$\eta_{1}q_{i_{1}}+\dots+\eta_{l}q_{i_{l}}+q_{e}+q_{e^{\prime}}=0.$

Thus,

$\displaystyle\varepsilon_{I}(e)$
$\displaystyle=\operatorname{sign}(\omega(q_{I}\star q_{e}))$
$\displaystyle=\operatorname{sign}(\omega(q_{I}\star(-\eta_{1}q_{i_{1}}-\dots-\eta_{l}q_{i_{l}}-q_{e^{\prime}})))$
$\displaystyle=\operatorname{sign}(\omega(q_{I}\star(-q_{e^{\prime}}))) $
$\displaystyle=-\varepsilon_{I}(e^{\prime}).$

The other case, with $e$ and $e^{\prime}$ having the same orientation, leads
to the opposite equality. Thus,
$\varepsilon_{F_{2},F_{1}}(e)\varepsilon_{I}(e)\in\\{-1,+1\\}$ does not depend
on the chosen $e$. Finally, we have proved (11). ∎

###  2.4. Symanzik rational fractions

In this subsection, we introduce the Symanzik rational fractions. The
following definition is more canonical that Definition 2.9 about Symanzik
polynomials. Indeed, the problems highlighted in Remarks 2.3 and 2.4 disappear
for Symanzik rational fractions. For instance, one can easily define Symanzik
rational fractions for families of vectors in $\mathbb{R}^{p}$. This is useful
in order to compute the archimedean height pairing, as we will see in
Subsection 3.3. Moreover, the rational fractions have a nice property of
stability stated in Theorem 5.1.

######  Definition 2.21.

Let $u$ be a family of vectors in $\mathbb{R}^{p}$, let $l$ be a nonnegative
integer and $w$ be a family of $l$ vectors in $\langle u\rangle$. The
_(normalized) Symanzik rational fraction of order $k$ of $u$ with parameters
$w_{1},\dots,w_{l}$_ is defined by

$\operatorname{\widetilde{\operatorname{Sym}}}_{k}(u;w;x):=\frac{\displaystyle\sum_{\begin{subarray}{c}I\subseteq[n]\\\
|I|=r-l\end{subarray}}\lVert\mathbf{u}_{I}\wedge\mathbf{w}\rVert^{k}x^{I^{c}}}{\displaystyle\sum_{\begin{subarray}{c}I\subseteq[n]\\\
|I|=r\end{subarray}}\lVert\mathbf{u}_{I}\rVert^{k}x^{I^{c}}}.\qed$

If elements of $u$ are in $\mathbb{Z}^{p}$, then we simply have

$\operatorname{\widetilde{\operatorname{Sym}}}_{k}(u;w;x)=\frac{\operatorname{Sym}_{k}(u;w;x)}{\operatorname{Sym}_{k}(u;x)}.$

Remark 3.18 gives a determinantal formula in the case $l=1$ and $k=2$. It can
be easily extended to the general case.

We have defined Kirchhoff and Symanzik polynomials for family of vectors, and
we have seen in Example 2.8 that they are linked to graphs. Thus, it is only
natural to look for a link with matroids.

###  2.5. The matroids case

This is a short subsection about the links between Kirchhoff and Symanzik
polynomials and matroids. More exactly, we give a definition of Kirchhoff and
Symanzik polynomials of order zero for matroids, and of any order for matroids
over hyperfields. This recent generalization of matroids by Baker and Bowler
can be found in [BB16]. To read this subsection, we assume that the reader is
familiar with (classical) matroids. Otherwise, the reader can go directly to
the next section, and a longer introduction to matroids will be given in
Section 4. All notions about matroids used here can be find in [Oxl11].

We will mainly use the axiomatic of independents. We write
$\mathfrak{M}=(E,\mathcal{I})$ for a matroid $\mathfrak{M}$ on a ground set
$E$ and with a family of independent sets
$\mathcal{I}\subseteq\operatorname{\mathcal{P}}(E)$.

If $u=(u_{1},\dots,u_{n})$ is a family of elements of a $\mathbb{Z}$-module,
$\mathfrak{M}_{u}=(E_{u},\mathcal{I}_{u})$, with $E_{u}=[n]$ will be the
_matroid representing the family $u$_.

We denote by $\operatorname{\mathcal{B}}(\mathfrak{M})$ the bases of
$\mathfrak{M}$ and by $\widehat{\mathfrak{M}}$ the dual of the matroid
$\mathfrak{M}$.

Let $n$ be a positive integer and $\mathfrak{M}=(E,\mathcal{I})$, with $E=[n]$
be a matroid. Here is the main definition.

######  Definition 2.22.

The _Kirchhoff polynomial (of order $0$) of the matroid $\mathfrak{M}$ with
variables $x_{1},\dots,x_{n}$_ is defined by

$\operatorname{Kir}_{0}(\mathfrak{M};x)=\sum_{I\in\operatorname{\mathcal{B}}(\mathfrak{M})}x^{I}.$

The _Symanzik polynomial (of order $0$) of the matroid $\mathfrak{M}$ with
variables $x_{1},\dots,x_{n}$_ is defined by

$\operatorname{Sym}_{0}(\mathfrak{M};x)=\sum_{I\in\operatorname{\mathcal{B}}(\widehat{\mathfrak{M}})}x^{I}.\qed$

This definition is natural because of the following claim, whose proof is
straightforward.

######  Claim 2.23.

Let $u$ be a family of $n$ elements in $\mathbb{Z}^{p}$. Then

$\displaystyle\operatorname{Kir}_{0}(u;x)=\operatorname{Kir}_{0}(\mathfrak{M}_{u};x),$
$\displaystyle\operatorname{Sym}_{0}(u;x)=\operatorname{Sym}_{0}(\mathfrak{M}_{u};x).$

The analogous for matroids of the duality Theorem 2.12 is given by the
following claim, which is a direct corollary of Claim 2.23 and of Theorem 2.12
for $k=0$.

######  Claim 2.24.

Let $q$ be a positive integer and $v$ be a family of $n$ vectors in
$\mathbb{Z}^{q}$ such that $v^{{}^{\intercal}}$ spans $\ker(U)$, then
$\mathfrak{M}_{u}=\widehat{\mathfrak{M}}_{v}$.

Next proposition shows that parameters of Symanzik polynomials correspond to a
so-called _quotient_ of the matroid.

If $w$ are elements of $\langle u\rangle$, then $\mathfrak{M}_{u\star w}$ is
an extension of $\mathfrak{M}_{u}$, i.e., we have $\mathfrak{M}_{u}$ equals
$\mathfrak{M}_{u\star w}\big{|}_{[n]}$, the induced matroid of
$\mathfrak{M}_{u\star w}$ on the elements $[n]$.

######  Proposition 2.25.

Let $w$ consist of independent elements of $\langle u\rangle$. Then

$\operatorname{Sym}_{0}(u;w;x)=\operatorname{Sym}_{0}(\mathfrak{M}_{u\star
w}\big{/}[n+1\,\raisebox{0.86108pt}{.}\,\raisebox{0.86108pt}{.}\,n+l];x).$

###### Proof.

Looking at Definition 2.16, the left-hand side is the sum of $x^{I^{c}}$ over
all $I\subseteq[n]$ of size $r-l$ such that $u_{I}\star w$ is an independent
family. But such sets $I$ exactly correspond to the bases of the matroid
$\mathfrak{M}_{u\star
w}\big{/}[n+1\,\raisebox{0.86108pt}{.}\,\raisebox{0.86108pt}{.}\,n+l]$. Thus,
Definition 2.22 ends the proof. ∎

Is there a “good” generalization of matroids suited to the computation of
Kirchhoff and Symanzik polynomials of higher order? Of course, one probably
has to assign some values to each basis of the matroid. In this regard,
matroids over hyperfields seem to be a good answer. They are far
generalizations of matroids given in [BB16]. Let $\mathbb{F}$ be a hyperfield
endowed with an involution $x\mapsto\overline{x}$. If $\mathfrak{M}$ is a
matroid over a $\mathbb{F}$ defined on the ground set $[n]$ by a Grassman-
Plücker function $\varphi$, then one can naturally define

$\displaystyle\operatorname{Kir}_{k}(\mathfrak{M};x)$
$\displaystyle:=\sum_{I\in\operatorname{\mathcal{B}}(\mathfrak{M})}\varphi(I)^{k}x^{I},$
$\displaystyle\operatorname{Sym}_{k}(\mathfrak{M};x)$
$\displaystyle:=\sum_{I\in\operatorname{\mathcal{B}}(\mathfrak{M})}\overline{\varphi(I)}^{k}x^{I^{c}},$
$\displaystyle\operatorname{Sym}_{k}(\mathfrak{M};w;x)$
$\displaystyle:=\operatorname{Sym}_{k}((\mathfrak{M}+w)/w;x),$

where $\mathfrak{M}+w$ is an extension of $\mathfrak{M}$. We still have the
duality

$\operatorname{Sym}_{k}(\mathfrak{M};x)=\operatorname{Kir}_{k}(\widehat{\mathfrak{M}};x),$

and the definitions coincide with ours for $\mathbb{R}$-matroids (up to a
factor). As observed in [BB16], duality for matroids over fields corresponds
to orthogonal complementation, as we have seen in Theorem 2.12.

##  3\. Symanzik polynomials on simplicial complexes

Every result in this section can be extended to CW-complexes.

Before extending Symanzik polynomials to simplicial complexes, we will
generalize the notion of forests in graph theory to the case of simplicial
complexes. Generalized forests will reveal interesting properties of these
polynomials.

###  3.1. Simplicial complexes and forests

Let $V$ be a finite set of _vertices_. An _abstract simplicial complex on $V$_
is a nonempty set $\Delta$ of subsets of $V$ called _faces_ such that $\Delta$
is stable by inclusion: if $\delta$ is a face and if $\gamma\subseteq\delta$,
then $\gamma$ is also a face. A simplicial complex $\Gamma$ is a _subcomplex_
of $\Delta$ if $\Gamma\subseteq\Delta$. If $\delta$ is a face, its _dimension_
is $\dim(\delta):=|\delta|-1$. Notice that $\Delta$ has always a unique face
of dimension $-1$: the empty set. The dimension of $\Delta$ is the maximal
dimension of its faces. We call it $d$. The $d$-dimensional faces are called
_facets_. If $l$ is an integer, $l\geqslant-1$, then $\Delta_{l}$ is the set
of faces of $\Delta$ of dimension $l$. The _$l$ -skeleton $\Delta_{(l)}$ of
$\Delta$_ is the subcomplex of all faces of dimension at most $l$ of $\Delta$:

$\Delta_{(l)}:=\bigcup_{i=-1}^{l}\Delta_{i}.$

In this article, we will suppose that a complex $\Delta$ is always endowed
with an enumeration on each set of faces $\Delta_{l}$,
$l\in[-1\,\raisebox{0.86108pt}{.}\,\raisebox{0.86108pt}{.}\,d]$, by numbers
from $1$ to $|\Delta_{l}|$.

Let $\operatorname{C}_{l}(\Delta):=\mathbb{Z}\langle\Delta_{l}\rangle$ be the
set of _$l$ -chains_ of $\Delta$, where each simplex has the standard
orientation given by the enumeration of $\Delta_{0}$. Let $\partial_{\Delta}$
be the _$d$ -th boundary operator_ of the augmented simplicial chain complex:

$\begin{array}[]{rrcl}\partial_{\Delta}:&\operatorname{C}_{d}(\Delta)&\longrightarrow&\operatorname{C}_{d-1}(\Delta),\\\
&\\{i_{0},\dots,i_{d}\\}\text{ with $i_{0}<\dots<i_{d}$
}&\longmapsto&\sum_{j=0}^{d}(-1)^{j}\\{i_{0},\dots,i_{j-1},i_{j+1},\dots,i_{d}\\}.\end{array}$

If $\delta_{i}$ denotes the $i$-th facet for the chosen enumeration, then
$(\delta_{1},\dots,\delta_{|\Delta_{d}|})$ is a basis of
$\operatorname{C}_{d}(\Delta)$. We identify $\operatorname{C}_{d}(\Delta)$
with $\mathbb{Z}^{|\Delta_{d}|}$ mapping this basis to the standard basis. We