# The tropical critical points of an affine matroid

Federico Ardila-Mantilla111San Francisco State University, Universidad de Los
Andes<EMAIL_ADDRESS>Supported by NSF Grant DMS-2154279.
Christopher Eur222Harvard University<EMAIL_ADDRESS>Supported by NSF
Grant DMS-2001854.
Raul Penaguiao333Max Planck Institute, Leipzig<EMAIL_ADDRESS>Supported by SNF Grant P2ZHP2 191301.

###### Abstract

We prove that the number of tropical critical points of an affine matroid
$(M,e)$ is equal to the beta invariant of $M$. Motivated by the computation of
maximum likelihood degrees, this number is defined to be the degree of the
intersection of the Bergman fan of $(M,e)$ and the inverted Bergman fan of
$N=(M/e)^{\perp}$, where $e$ is an element of $M$ that is neither a loop nor a
coloop. Equivalently, for a generic weight vector $w$ on $E-e$, this is the
number of ways to find weights $(0,x)$ on $M$ and $y$ on $N$ with $x+y=w$ such
that on each circuit of $M$ (resp. $N$), the minimum $x$-weight (resp.
$y$-weight) occurs at least twice. This answers a question of Sturmfels.

##  1 Introduction

During the Workshop on Nonlinear Algebra and Combinatorics from Physics at the
Center for the Mathematical Sciences and Applications at Harvard University in
April 2022, Bernd Sturmfels [Stu22] posed one of those combinatorial problems
that is deceivingly simple to state, but whose answer requires a deeper
understanding of the objects at hand.

######  Conjecture 1.1.

[Stu22] Let $M$ be a matroid on $E$, and let $e\in E$ be an element that is
neither a loop nor a coloop. Let $M/e$ be the contraction of $M$ by $e$ and
let $N=(M/e)^{\perp}$ be its dual matroid.

  1. 1.

_(Combinatorial version)_ Given a vector $\mathsf{w}\in\mathbb{R}^{E-e}$, we
wish to find weight vectors $(0,\mathsf{x})\in\mathbb{R}^{E}$ on $M$ (where
$e$ has weight $0$) and $\mathsf{y}\in\mathbb{R}^{E-e}$ on $N$ such that

$\bullet$ on each circuit of $M$, the minimum $\mathsf{x}$-weight occurs at
least twice,

$\bullet$ on each circuit of $N$, the minimum $\mathsf{y}$-weight occurs at
least twice, and

$\bullet$ $\mathsf{w}=\mathsf{x}+\mathsf{y}$.

For generic $\mathsf{w}$, the number of solutions is the beta invariant
$\beta(M)$.

  2. 2.

_(Geometric version)_ The degree of the stable intersection of the Bergman fan
$\Sigma_{(M,e)}$ and the inverted Bergman fan $-\Sigma_{N}$ is

$\deg(\Sigma_{(M,e)}\cdot-\Sigma_{(M/e)^{\perp}})=\beta(M)\,.$

The goal of this paper is to prove this conjecture.

######  Theorem 1.2.

Versions 1. and 2. of 1.1 are true.

The affine matroid $(M,e)$ is the matroid $M$ with a special chosen element
$e$. The Bergman fan of $(M,e)$ is the Bergman fan of $M$ intersected with the
hyperplane $x_{e}=0$. The other relevant definitions are given in Section 2.3.
The combinatorial and geometric formulations of 1.1 are equivalent because in
the stable intersection above, all intersection points have multiplicity 1
[ABF+21, Lemma 7.4].

Figure 1: A graph $G$, its contraction $G/0$, and its dual $H=(G/0)^{\perp}$.

Agostini, Brysiewicz, Fevola, Kühne, Sturmfels, and Telen [ABF+21] first
encountered (a special case of) 1.1 in their study of the maximum likelihood
estimation for linear discrete models. Using algebro-geometric results of Huh
and Sturmfels [HS14], which built on earlier work of Varchenko [Var95], they
proved 1.2 for matroids realizable over the real numbers. In a related setting
of linear Gaussian models, the maximum likelihood degrees were shown to be
matroid invariants of the linear subspace [SU10, EFSS21].

We prove 1.2 for all matroids. Following the original motivation, we call the
solutions to 1.1.1 the _tropical critical points_ of the affine matroid; our
main result is that they are counted by Crapo’s beta invariant $\beta(M)$. We
do something stronger. Agostini et. al. write

> “we would like to describe the multivalued map that takes any tropical data
> vector $\mathsf{w}$ to the set of its critical points”. [ABF+21, Section 7]

We give an explicit formula for this map for all $\mathsf{w}$ that are rapidly
increasing, under any order $<$ on the ground set $E$.

In Section 3 we prove 1.2.1 combinatorially, relying on the tropical geometric
fact that the number of solutions is the same for all generic $\mathsf{w}$. We
show that when the entries of $\mathsf{w}$ are rapidly increasing with respect
to some order $<$ on $E$, the solutions to 1.1.1 are naturally in bijection
with the $\beta$-nbc bases of the matroid with respect to $<$. It is known
that the number of such bases is the beta invariant of the matroid, regardless
of the order $<$.

In Section 4 we sketch a proof of 1.2.2 that relies the theory of tautological
classes of matroids of Berget, Eur, Spink, and Tseng [BEST21].

##  2 Notation and preliminaries

###  2.1 The lattice of set partitions

A set partition $\boldsymbol{\lambda}$ of a set $E$ is a collection of
subsets, called blocks, of $E$, say
$\boldsymbol{\lambda}=\\{\lambda_{1},\dots,\lambda_{\ell}\\}$, whose union is
$E$ and whose pairwise intersections are empty. We write
$\boldsymbol{\lambda}\models E$. We let $|\boldsymbol{\lambda}|=\ell$ be the
number of blocks of $\boldsymbol{\lambda}$. If $e\in E$ and
$\boldsymbol{\lambda}\models E$, we write $\boldsymbol{\lambda}(e)$ for the
block of $\boldsymbol{\lambda}$ that contains $e$.

We define the _linear space of a set partition
$\boldsymbol{\lambda}=\\{\lambda_{1},\ldots,\lambda_{\ell}\\}\models E$_ to be

$\displaystyle\mathsf{L}(\boldsymbol{\lambda})$ $\displaystyle\coloneqq$
$\displaystyle\textrm{span}\\{{\mathsf{e}}_{\lambda_{1}},\ldots,{\mathsf{e}}_{\lambda_{\ell}}\\}\,\subseteq\mathbb{R}^{E}$
$\displaystyle=$
$\displaystyle\\{\mathsf{x}\in\mathbb{R}^{E}\,\,|\,x_{i}=x_{j}\text{ whenever
}i,j\text{ are in the same block of }\boldsymbol{\lambda}\\},$

where $\\{\mathsf{e}_{i}\,:\,i\in E\\}$ is the standard basis of
$\mathbb{R}^{E}$ and $\mathsf{e}_{S}=\sum_{s\in S}e_{s}$ for $S\subseteq E$.
Notice that $\dim\mathsf{L}(\boldsymbol{\lambda})=|\boldsymbol{\lambda}|$. The
map $\boldsymbol{\lambda}\mapsto\mathsf{L}(\boldsymbol{\lambda})$ is a
bijection between the set partitions of $E$ and the flats of the _braid
arrangement_ , which is the hyperplane arrangement in $\mathbb{R}^{E}$ given
by the hyperplanes $x_{i}=x_{j}$ for $i\neq j$ in $E$.

If $e\in E$ then we write
$\mathsf{L}(\boldsymbol{\lambda})|_{x_{e}=0}=\\{\mathsf{x}\in\mathbb{R}^{E-e}\,:\,(0,\mathsf{x})\in\mathsf{L}(\boldsymbol{\lambda})\subseteq\mathbb{R}^{E}\\}$.

###  2.2 The intersection graph of two set partitions

The following construction from [AE21] will play an important role.

######  Definition 2.1.

Let $\boldsymbol{\lambda}\models[0,n]$ and $\boldsymbol{\mu}\models[n]$ be set
partitions. The _intersection graph_
$\Gamma=\Gamma_{\boldsymbol{\lambda},\boldsymbol{\mu}}$ is the bipartite graph
with vertex set $\boldsymbol{\lambda}\sqcup\boldsymbol{\mu}$ and edge set
$[n]$, where the edge labelled $e$ connects the parts $\lambda(e)$ of
$\boldsymbol{\lambda}$ and $\mu(e)$ of $\boldsymbol{\mu}$ containing $e$. The
vertex $\lambda(0)$ is marked with a hollow point.

Figure 2: The intersection graph of
$\boldsymbol{\lambda}=\\{6,59,2,013478\\}\models[0,9]$ and
$\boldsymbol{\mu}=\\{9,8,7,46,3,125\\}\models[9]$. We omit brackets for
legibility. Left: The vertices are labelled by the blocks of the set
partitions. Right: The edges are labelled by the elements of $[9]$.

The intersection graph may have several parallel edges connecting the same
pair of vertices. Notice that the label of a vertex in $\Gamma$ is just the
set of labels of the edges incident to it. Therefore we can remove the vertex
labels, and simply think of $\Gamma$ as a bipartite multigraph on edge set
$[n]$. This is illustrated in Figure 2.

######  Lemma 2.2.

Let $\boldsymbol{\lambda}\models[0,n]$ and $\boldsymbol{\mu}\models[n]$ be set
partitions and $\Gamma_{\boldsymbol{\lambda},\boldsymbol{\mu}}$ be their
intersection graph.

  1. 1.

If $\Gamma_{\boldsymbol{\lambda},\boldsymbol{\mu}}$ has a cycle, then
$\mathsf{L}(\boldsymbol{\lambda})|_{x_{0}=0}\cap(\mathsf{w}-\mathsf{L}(\boldsymbol{\mu}))=\emptyset$
for generic444This means that this property holds for all $\mathsf{w}$ outside
of a set of measure $0$. $\mathsf{w}\in\mathbb{R}^{n}$.

  2. 2.

If $\Gamma_{\boldsymbol{\lambda},\boldsymbol{\mu}}$ is disconnected, then
$\mathsf{L}(\boldsymbol{\lambda})|_{x_{0}=0}\cap(\mathsf{w}-\mathsf{L}(\boldsymbol{\mu}))$
is not a point for any $\mathsf{w}\in\mathbb{R}^{n}$.

  3. 3.

If $\Gamma_{\boldsymbol{\lambda},\boldsymbol{\mu}}$ is a tree, then
$\mathsf{L}(\boldsymbol{\lambda})|_{x_{0}=0}\cap(\mathsf{w}-\mathsf{L}(\boldsymbol{\mu}))$
is a point for any $\mathsf{w}\in\mathbb{R}^{n}$.

###### Proof.

Let $\mathsf{x}\in\mathsf{L}(\boldsymbol{\lambda})$ and
$\mathsf{y}\in\mathsf{L}(\mu)$ such that $\mathsf{x}+\mathsf{y}=\mathsf{w}$.
Write $x_{\lambda(i)}\coloneqq x_{i}$ and $y_{\mu(j)}\coloneqq y_{j}$ for
simplicity. The subspace
$\mathsf{L}(\boldsymbol{\lambda})|_{x_{0}=0}\cap(\mathsf{w}-\mathsf{L}(\boldsymbol{\mu}))$
is cut out by the equalities

$\displaystyle x_{\lambda(i)}+y_{\mu(i)}$ $\displaystyle=$ $\displaystyle
w_{i}\qquad\textrm{ for }i\in[n],$ $\displaystyle x_{\lambda(0)}$
$\displaystyle=$ $\displaystyle 0.$

This system has $n+1$ equations and
$|\boldsymbol{\lambda}|+|\boldsymbol{\mu}|$ unknowns. The linear dependences
among these equations are controlled by the cycles of the graph
$\Gamma_{\boldsymbol{\lambda},\boldsymbol{\mu}}$. More precisely, the first
$|E|$ linear functionals $\\{x_{\lambda(i)}+y_{\mu(i)}\,:\,i\in[n]\\}$ give a
realization of the graphical matroid of
$\Gamma_{\boldsymbol{\lambda},\boldsymbol{\mu}}$. The last equation is clearly
linearly independent from the others.

If $\Gamma_{\boldsymbol{\lambda},\boldsymbol{\mu}}$ has a cycle with edges
$i_{1},i_{2},\ldots,i_{2k}$ in that order, then the above equalities imply
that $w_{i_{1}}-w_{i_{2}}+w_{i_{3}}-\cdots-w_{i_{2k}}=0$. For a generic
$\mathsf{w}$, this equation does not hold, so we have
$\mathsf{L}(\boldsymbol{\lambda})|_{x_{0}=0}\cap(\mathsf{w}-\mathsf{L}(\boldsymbol{\mu}))=\emptyset$.

If $\Gamma_{\boldsymbol{\lambda},\boldsymbol{\mu}}$ is disconnected, let $A$
be the set of edges in a connected component not containing the vertex
$\lambda(0)$. If $\mathsf{x}\in\mathsf{L}(\boldsymbol{\lambda})$ and
$\mathsf{y}\in\mathsf{L}(\boldsymbol{\mu})$ satisfy
$\mathsf{x}+\mathsf{y}=\mathsf{w}$ and $x_{0}=0$, then
$\mathsf{x}+r\mathsf{e}_{A}\in\mathsf{L}(\boldsymbol{\lambda})$ and
$\mathsf{y}-r\mathsf{e}_{A}\in\mathsf{L}(\boldsymbol{\mu})$ also satisfy those
equations for any real number $r$. Therefore
$\mathsf{L}(\boldsymbol{\lambda})|_{x_{0}=0}\cap(\mathsf{w}-\mathsf{L}(\boldsymbol{\mu}))$
is not a point.

Finally, if $\Gamma_{\boldsymbol{\lambda},\boldsymbol{\mu}}$ is a tree, then
its number of vertices is one more than the number of edges, that is,
$n+1=|\boldsymbol{\lambda}|+|\boldsymbol{\mu}|$, so the system of equations
has equally many equations and unknowns. Also, these equations are linearly
independent since $\Gamma_{\boldsymbol{\lambda},\boldsymbol{\mu}}$ is a tree.
It follows that the system has a unique solution. ∎

When $\Gamma_{\boldsymbol{\lambda},\boldsymbol{\mu}}$ is a tree, we call
$\boldsymbol{\lambda}$ and $\boldsymbol{\mu}$ an _arboreal pair_.

######  Lemma 2.3.

Let $\boldsymbol{\lambda}\models[0,n]$ and $\boldsymbol{\mu}\models[n]$ be an
arboreal pair of set partitions and let
$\Gamma_{\boldsymbol{\lambda},\boldsymbol{\mu}}$ be their intersection tree.
Let $\mathsf{w}\in\mathbb{R}^{n}$. The unique vectors
$\mathsf{x}\in\mathsf{L}(\boldsymbol{\lambda})$ and
$\mathsf{y}\in\mathsf{L}(\boldsymbol{\mu})$ such that
$\mathsf{x}+\mathsf{y}=\mathsf{w}$ and $x_{0}=0$ are given by

$\displaystyle x_{\lambda_{i}}$ $\displaystyle=$ $\displaystyle
w_{e_{1}}-w_{e_{2}}+\cdots\pm w_{e_{k}}\qquad\textrm{ where $e_{1}e_{2}\ldots
e_{k}$ is the unique path from $\lambda_{i}$ to $\lambda(0)$}$ $\displaystyle
y_{\mu_{j}}$ $\displaystyle=$ $\displaystyle w_{f_{1}}-w_{f_{2}}+\cdots\pm
w_{f_{l}}\qquad\textrm{ where $f_{1}f_{2}\ldots f_{l}$ is the unique path from
$\mu_{j}$ to $\lambda(0)$}$

for any $i$ and $j$.

###### Proof.

This follows readily from the fact that, for each $1\leq i\leq k$, the values
of $x_{\lambda(e_{i})}$ and $y_{\mu(e_{i})}$ on the vertices incident to edge
$i$ have to add up to $w_{e_{i}}$. ∎

######  Example 2.4.

Let $\boldsymbol{\lambda}=\\{6,59,2,013478\\}\models[0,9]$ and
$\boldsymbol{\mu}=\\{9,8,7,46,3,125\\}\models~{}[9]$. These set partitions
form an arboreal pair, as evidenced by their intersection tree, shown in
Figure 2. We have, for example, $y_{9}=w_{9}-w_{5}+w_{1}$ because the path
from $\mu(9)=\\{9\\}$ to $\lambda(0)=\\{013478\\}$ uses edges $9,5,1$ in that
order. The remaining values are:

$\displaystyle x_{6}=w_{6}-w_{4},\quad x_{59}=w_{5}-w_{1},\quad
x_{2}=w_{2}-w_{1},\quad x_{13478}=0,$ $\displaystyle
y_{9}=w_{9}-w_{5}+w_{1},\quad y_{8}=w_{8},\quad y_{7}=w_{7},\quad
y_{46}=w_{4},\quad y_{3}=w_{3},\quad y_{1235}=w_{1}.$

The tropical critical points of a matroid are better behaved for the following
family of vectors.

######  Definition 2.5.

A vector $\mathsf{w}\in\mathbb{R}^{n+1}$ is _rapidly increasing_ if
$w_{i+1}>3w_{i}>0$ for $1\leq i\leq n$.

The next lemma is readily verified.

######  Lemma 2.6.

Let $\mathsf{w}$ be rapidly increasing. For any $1\leq a<b\leq n+1$ and any
choice of $\epsilon_{i}$s and $\delta_{i}$s in $\\{-1,0,1\\}$, we have
$w_{a}+\sum_{i=1}^{a-1}\epsilon_{i}w_{i}<w_{b}+\sum_{j=1}^{b-1}\delta_{j}w_{j}.$

######  Definition 2.7.

Given a rapidly increasing vector $\mathsf{w}\in\mathbb{R}^{n+1}$ and a real
number $x$, we will say $x$ _is near_ $w_{i}$ and write $x\approx w_{i}$ if
$w_{i}-(w_{1}+\cdots+w_{i-1})\leq x\leq w_{i}+(w_{1}+\cdots+w_{i-1})$. Note
that if $x\approx w_{i}$ and $y\approx w_{j}$ for $i<j$ then $x<y$.

###  2.3 Matroids, Bergman fans, and tropical geometry

In what follows we will assume familiarity with basic notions in matroid
theory; for definitions and proofs, see [Oxl06, Wel76]. We also state here
some facts from tropical geometry that we will need; see [MS15, MR10] for a
thorough introduction.

Let $M$ be a matroid on $E$ of rank $r+1$. The _dual matroid_ $M^{\perp}$ is
the matroid on $E$ whose set of bases is $\\{B^{\perp}\,|\,B\text{ is a basis
of $M$}\\}$, where $B^{\perp}\coloneqq E-B$. The following lemma is useful to
how $M$ and $M^{\perp}$ interact; see [ADH22, Lemma 3.14] and [Oxl06,
Proposition 2.1.11] for proofs.

######  Lemma 2.8.

If $F$ is a flat of $M$ and $G$ is a flat of $M^{\perp}$, then $|F\cup
G|\neq|E|-1$.

######  Definition 2.9.

[Cra67] The _beta invariant_ of $M$ is defined to be
$\beta(M)\coloneqq|\chi_{M}^{\prime}(1)|$, where $\chi_{M}$ is the
_characteristic polynomial_ of $M$:

$\chi_{M}(t)\coloneqq\sum_{X\subseteq E}(-1)^{|X|}t^{r(M)-r(X)}.$

######  Definition 2.10.

Fix a linear order $<$ on $M$. A _broken circuit_ is a set of the form
$C-\text{min}_{<}C$ where $C$ is a circuit of $M$. An _nbc-basis_ of $M$ is a
basis of $M$ that contains no broken circuits. A _$\beta$ nbc-basis_ of $M$ is
an nbc-basis $B$ such that $B^{\perp}\cup 0\setminus 1$ is an nbc-basis of
$M^{\perp}$.

######  Theorem 2.11.

[Zie92] For any linear order $<$ on $E$, the number of $\beta$nbc-bases of $M$
is equal to the beta invariant $\beta(M)$.

For each basis $B=\\{b_{1}>\dots>b_{r}>b_{r+1}\\}$ of the matroid $M$, we
define the complete flag of flats

$\mathcal{F}_{M}(B)\coloneqq\\{\emptyset\subsetneq\mathrm{cl}_{M}\\{b_{1}\\}\subsetneq\mathrm{cl}_{M}\\{b_{1},b_{2}\\}\subsetneq\cdots\subsetneq\mathrm{cl}_{M}\\{b_{1},\dots,b_{r}\\}\subsetneq
E\\}.$

The following characterization of nbc-bases will be useful.

######  Lemma 2.12.

[Bjö92, (7.30), (7.31)] Let $M$ be a matroid of size $n+1$ and rank $r+1$, and
$B$ a basis of $M$. Then $B$ is an nbc-basis of $M$ if and only if $b_{i}=\min
F_{i}$ for $i=1,\dots,r+1$.

An _affine matroid_ $(M,e)$ on $E$ is a matroid $M$ on $E$ with a chosen
element $e\in E$ [Zie92].

######  Definition 2.13.

[Stu02] The _Bergman fan_ of a matroid $M$ on $E$ is

$\Sigma_{M}=\\{\mathsf{x}\in\mathbb{R}^{E}\,|\,\min_{c\in C}x_{c}\text{ is
attained at least twice for any circuit $C$ of $M$}\\}\,.$

The _Bergman fan of an affine matroid_ $(M,e)$ on $E$ is

$\Sigma_{(M,e)}=\\{\mathsf{x}\in\mathbb{R}^{E-e}\,|\,(0,\mathsf{x})\in\Sigma_{M}\\}=\Sigma_{M}|_{x_{e}=0}.$

######  Remark 2.14.

The Bergman fan contains the lineality space $\mathbbm{1}\mathbb{R}$. Taking
the quotient by this space, or intersecting with a coordinate linear
hyperplane will give the same result, and typically the (projective) Bergman
fan is defined in the quotient vector space
$\mathbb{R}^{E}/{\mathbbm{1}\mathbb{R}}$ in the literature.

The motivation for this definition comes from tropical geometry. A subspace
$V\subset\mathbb{R}^{E}$ determines a matroid $M_{V}$ on $E$, and the
tropicalization of $V$ is precisely the Bergman fan of $M_{V}$. Similarly, an
affine subspace $W\subset\mathbb{R}^{E-e}$ determines an affine matroid
$(M_{W},e)$ on $E$, where $e$ represents the hyperplane at infinity. The
tropicalization of $W$ is the Bergman fan $\Sigma_{(M_{W},e)}$.

######  Theorem 2.15.

[AK06] The Bergman fan of a matroid $M$ is equal to the union of the cones

$\displaystyle\sigma_{\mathcal{F}}$ $\displaystyle=$
$\displaystyle\textrm{cone}(\mathsf{e}_{F_{1}},\ldots,\mathsf{e}_{F_{r+1}})+\mathbb{R}\mathbbm{1}$
$\displaystyle=$ $\displaystyle\\{\mathsf{x}\in\mathbb{R}^{E}\,|\,x_{a}\geq
x_{b}\text{ whenever $a\in F_{i}$ and $b\in F_{j}$ for some $1\leq i\leq j\leq
r+1$}\\}$

for the complete flags $\mathcal{F}=\\{\emptyset=F_{0}\subsetneq
F_{1}\subsetneq\dots\subsetneq F_{r}\subsetneq F_{r+1}=E\\}$ of flats of $M$.
It is a tropical fan with weights $w(\mathcal{F})=1$ for all $\mathcal{F}$.

If $\Sigma_{1}$ and $\Sigma_{2}$ are tropical fans of complementary
dimensions, then $\Sigma_{1}$ and $v+\Sigma_{2}$ intersect transversally at a
finite set of points for any sufficiently generic vector $v\in\mathbb{R}^{n}$.
Furthermore, each intersection point $p$ is equipped with a multiplicity
$w(p)$ that depends on the respective intersecting cones, in such a way that
the quantity

$\deg(\Sigma_{1}\cdot\Sigma_{2}):=\sum_{p\in\Sigma_{1}\cap(v+\Sigma_{2})}w(p)$

is constant for generic $v$ [MR10, Proposition 4.3.3, 4.3.6]; this is called
the _degree_ of the intersection.

In all the tropical intersections that arise in this paper, it was verified in
[ABF+21, Lemma 7.4] that the multiplicity index $\omega(p)$ is 1. This also
follows readily from the fact that every such intersection comes from an
arboreal pair $\boldsymbol{\lambda}$, $\boldsymbol{\mu}$ by 2.2, as explained
in the next section. Therefore the degree of the intersection will be simply
the number of intersection points:

$\deg(\Sigma_{(M,e)}\cdot-\Sigma_{(M/e)^{\perp}})=|\Sigma_{(M,e)}\cap(v-\Sigma_{(M/e)^{\perp}})|$

for generic $v\in\mathbb{R}^{E-e}$. This explains the equivalence of the two
versions of 1.1 and 1.2.

##  3 Proof of the main theorem via basis activities

Let $M$ be a matroid on $[0,n]$ of rank $r+1$ such that $0$ is not a loop nor
a coloop. Then $M/0$ has rank $r$, and $N=(M/0)^{\perp}$ has rank $n-r$. For
any basis $B$ of $M$ containing $0$, $B^{\perp}=[0,n]-B$ is a basis of
$N=(M/0)^{\perp}$. Conversely, every basis of $N$ equals $B^{\perp}$ for a
basis $B$ of $M$ containing $0$.

Let us construct an intersection point in
$\Sigma_{(M,0)}\cap(\mathsf{w}-\Sigma_{N})$ for each $\beta$-nbc basis of $M$.

######  Lemma 3.1.

Let $M$ be a matroid on $E=[0,n]$ of rank $r+1$ such that $0$ is not a coloop,
and let $N=(M/0)^{\perp}$. Let $\mathsf{w}\in\mathbb{R}^{n}$ be rapidly
increasing. For any $\beta$-nbc basis $B$ of $M$, there exist unique vectors
$(0,\mathsf{x})\in\sigma_{\mathcal{F}_{M}(B)}$ and
$\mathsf{y}\in\sigma_{\mathcal{F}_{N}(B^{\perp})}$ such that
$\mathsf{x}+\mathsf{y}=\mathsf{w}$.

###### Proof.

First we show that the set partitions ${\boldsymbol{\pi}}$ of
$\mathcal{F}=\mathcal{F}_{M}(B)$ and ${\boldsymbol{\pi}}^{\perp}$ of
$\mathcal{F}^{\perp}=\mathcal{F}_{N}(B^{\perp})$ form an arboreal pair. Since
they have sizes $|B|=r+1$ and $|B^{\perp}|=n-r$, respectively, their
intersection graph has $n+1$ vertices and $n$ edges. Therefore it is
sufficient to prove that the intersection graph
$\Gamma_{{\boldsymbol{\pi}},{\boldsymbol{\pi}}^{\perp}}$ is connected; this
implies that it is a tree.

Assume contrariwise, and let $A$ be a connected component not containing the
edge $1$. Let $a>1$ be the smallest edge in $A$. Then $a$ is the smallest
element of its part ${\boldsymbol{\pi}}(a)$ in ${\boldsymbol{\pi}}$, and since
$B$ is nbc in $M$, this implies $a\in B$. Similarly, since $B^{\perp}$ is nbc
in $N$, this also implies $a\in B^{\perp}$. This is a contradiction.

It follows from Lemma 2.2 that there exist unique
$(0,\mathsf{x})\in\mathsf{L}(\boldsymbol{\pi})$ and
$\mathsf{y}\in\mathsf{L}(\boldsymbol{\pi}^{\perp})$ such that
$\mathsf{x}+\mathsf{y}=\mathsf{w}$. It remains to show that
$(0,\mathsf{x})\in\sigma_{\mathcal{F}}$ and
$\mathsf{y}\in\sigma_{\mathcal{F}^{\perp}}$.

Lemma 2.3 provides formulas for $\mathsf{x}$ and $\mathsf{y}$ in terms of the
paths from the various vertices of the tree of
$\Gamma_{\boldsymbol{\pi},\boldsymbol{\pi}^{\perp}}$ to $\pi(0)$. To
understand those paths, let us give each edge $e$ an orientation as follows:

$\displaystyle\pi(e)\longrightarrow\pi^{\perp}(e)$ if
$\displaystyle\min\pi(e)>\min\pi^{\perp}(e),$
$\displaystyle\pi(e)\longleftarrow\pi^{\perp}(e)$ if
$\displaystyle\min\pi(e)<\min\pi^{\perp}(e).$

We never have $\min\pi(e)=\min\pi^{\perp}(e)$, because as above, that would
imply $e\in B\cap B^{\perp}$.

We claim that every vertex other than $\pi(0)$ has an outgoing edge under this
orientation. Consider a part $\pi_{i}\neq\pi(0)$ of ${\boldsymbol{\pi}}$; let
$\min\pi_{i}=b$. Edge $b$ connects $\pi_{i}=\pi(b)$ to $\pi^{\perp}(b)\ni b$,
and we cannot have $\min\pi^{\perp}(b)>b=\min\pi(b)$, so we must have
$\pi_{i}\rightarrow\pi^{\perp}(b)$. The same argument works for any part
$\pi^{\perp}_{j}$ of ${\boldsymbol{\pi}}^{\perp}$.

Now, since $B$ is an nbc basis of $M$, every element $b\in B$ is minimum in
$\pi(b)$, so there is a directed path that starts at $\pi(b)$ and can only end
at $\pi(0)$, and its first edge is $b$. Furthermore, by the definition of the
orientation, the labels of the edges decrease along this path. Thus in the
alternating sum $x_{b}=w_{b}\pm\cdots$ given by Lemma 2.3, the first term
dominates, and $x_{b}\approx w_{b}$. Similarly, since $B^{\perp}$ is an nbc
basis of $N$, $y_{c}\approx w_{c}$ for all $c\in B^{\perp}$.

Therefore, if we write $B=\\{b_{1}>\dots>b_{r}>b_{r+1}=0\\}$, since $w$ is
rapidly increasing, it follows that
$x_{b_{1}}>x_{b_{2}}>\cdots>x_{b_{r}}>x_{b_{r+1}}=0$, so indeed
$(0,\mathsf{x})\in\sigma_{\mathcal{F}}$. Similarly, if we write
$B^{\perp}=E-B=\\{c_{1}>\dots>c_{n-r}>c_{n-r+1}=1\\}$, then
$y_{c_{1}}>y_{c_{2}}>\cdots>y_{c_{n-r+1}}$, so
$\mathsf{y}\in\sigma_{\mathcal{F}^{\perp}}$. The desired result follows. ∎

######  Example 3.2.

The graphical matroid $M$ of the graph $G$ in Figure 1 has six $\beta$-nbc
bases: $0256$, $0257$, $0259$, $0368$, $0378$, $0379$. Let us compute the
intersection point in $\Sigma_{(M,0)}\cap(\mathsf{w}-\Sigma_{N})$ associated
to $0257$ for the rapidly increasing vector
$\mathsf{w}=(10^{0},10^{1},\ldots,10^{8})\in\mathbb{R}^{9}$.

For $B=0257$, we have $B^{\perp}=134689$. The flags they generate in $M$ and
$N$ are

$\displaystyle\mathcal{F}_{M}(B)$ $\displaystyle=$
$\displaystyle\\{\emptyset\subsetneq 7\subsetneq 57\subsetneq 2457\subsetneq
0123456789\\}$ $\displaystyle\mathcal{F}_{N}(B^{\perp})$ $\displaystyle=$
$\displaystyle\\{\emptyset\subsetneq 9\subsetneq 89\subsetneq 689\subsetneq
46789\subsetneq 346789\subsetneq 123456789\\},$

which give rise to the corresponding set compositions

${\boldsymbol{\pi}}=7|5|24|013689,\qquad{\boldsymbol{\pi}}^{\perp}=9|8|6|47|3|125.$

This is indeed an arboreal pair, as evidenced by their intersection graph in
Figure 3.

Figure 3: The intersection graph of ${\boldsymbol{\pi}}=7|5|24|13689$ and
${\boldsymbol{\pi}}^{\perp}=9|8|6|47|3|125$.

3.1 gives us the unique points
$(0,\mathsf{x})\in\mathcal{F}_{{\boldsymbol{\pi}}}$ and
$\mathsf{y}\in\mathcal{F}_{{\boldsymbol{\tau}}}$ such that
$\mathsf{x}+\mathsf{y}=\mathsf{w}$; they are given by the paths to the special
vertex $\pi(0)$ in the intersection tree
$\Gamma_{{\boldsymbol{\pi}},{\boldsymbol{\pi}}^{\perp}}$. For example
$x_{7}=10^{6}-10^{3}+10^{1}-10^{0}=999009$ and
$y_{4}=10^{3}-10^{1}+10^{0}=991$ are given by the paths 7421 and 421 from
$\pi(7)=\pi_{1}$ and $\pi^{\perp}(7)=\pi^{\perp}_{4}$ to $\pi(0)$,
respectively. In this way we obtain:

$\begin{array}[]{rrrrrrrrrr}\mathsf{x}=&0&9&0&9&9999&0&999009&0&0\\\
\mathsf{y}=&1&1&100&991&1&100000&991&10000000&100000000\\\
\mathsf{w}=&1&10&100&1000&10000&100000&1000000&10000000&100000000\end{array}$

and $\mathsf{x}\in\Sigma_{(M,0)}\cap(\mathsf{w}-\Sigma_{N})$. We invite the
reader to record the weights $(0,\mathsf{x})$ and $\mathsf{y}$ in the graphs
$G$ and $H$ of Figure 1, and verify that in each cycle the minimum weight
appears at least twice.

Conversely, the following lemma shows that any intersection point between
$\Sigma_{(E,e)}$ and $v-\Sigma_{N}$ is of the form constructed in 3.1; that
is, it comes from a $\beta$-nbc basis.

######  Lemma 3.3.

Let $M$ be a matroid on $E=[0,n]$ of rank $r+1$, such that $0$ is not a loop
nor a coloop, and $N=(M/0)^{\perp}$. Let $\mathsf{w}\in\mathbb{R}^{n}$ be
generic and rapidly increasing. Let

$\displaystyle\mathcal{F}$ $\displaystyle=$
$\displaystyle\\{\emptyset=F_{0}\subsetneq F_{1}\subsetneq\dots\subsetneq
F_{r}\subsetneq F_{r+1}=E\\}$ $\displaystyle\mathcal{G}$ $\displaystyle=$
$\displaystyle\\{\emptyset=G_{0}\subsetneq G_{1}\subsetneq\dots\subsetneq
G_{n-r-1}\subsetneq G_{n-r}=E-0\\}$

be complete flags of the matroids $M$ and $N$, respectively, such that
$\Sigma_{(M,0)}$ and $\mathsf{w}-\Sigma_{N}$ intersect at
$\sigma_{\mathcal{F}}$ and $\mathsf{w}-\sigma_{\mathcal{G}}$. Then there
exists a $\beta$-nbc basis $B$ of $M$ such that
$\mathcal{F}=\mathcal{F}_{M}(B)$ and $\mathcal{G}=\mathcal{F}_{N}(B^{\perp})$.

###### Proof.

By Lemma 2.2, the set compositions ${\boldsymbol{\pi}}$ and
${\boldsymbol{\tau}}$ of $\mathcal{F}$ and $\mathcal{G}$ form an arboreal
pair. In particular, $\pi_{a}\cap\tau_{b}=(F_{a}-F_{a-1})\cap(G_{b}-G_{b-1})$
cannot have more than one element for any $a$ and $b$. We proceed in several
steps.

1\. Our first step will be to show that in the intersection tree
$\Gamma_{{\boldsymbol{\pi}},{\boldsymbol{\tau}}}$, the top right vertex
$\pi_{r+1}$ contains $0$ and $1$, the bottom right vertex $\tau_{n-r}$
contains $1$, and thus the edge $1$ connects these two rightmost vertices.

Each $G_{i}$ is a flat of $N=M^{\perp}-0$, so
$G_{i}^{\bullet}\coloneqq\mathrm{cl}_{M^{\perp}}(G_{i})\in\\{G_{i},G_{i}\cup
0\\}$ is a flat of $M^{\perp}$. Consider the flag of flats of $M^{\perp}$

$\mathcal{G}^{\bullet}\coloneqq\\{\emptyset=G_{0}^{\bullet}\subsetneq
G_{1}^{\bullet}\subsetneq\dots\subsetneq G_{n-r-1}^{\bullet}\subsetneq
G_{n-r}^{\bullet}=E\\},$

where $G_{n-r}^{\bullet}=E$ because $0$ is not a coloop of $M^{\perp}$ and
$G_{0}^{\bullet}=\emptyset$ because $0$ is not a loop of $M^{\perp}$. Let $m$
be the minimal index such that $0\in G_{m}^{\bullet}$, so

$\mathcal{G}^{\bullet}\coloneqq\\{\emptyset=G_{0}\subsetneq
G_{1}\subsetneq\dots\subsetneq G_{m-1}\subsetneq G_{m}\cup
0\subsetneq\dots\subsetneq G_{n-r-1}\cup 0\subsetneq G_{n-r}\cup 0=E\\},$

Consider the unions of the flat $F_{r}$ with the coflats in
$\mathcal{G}^{\bullet}$; let $j$ be the index such that

$F_{r}\cup G_{j-1}^{\bullet}\neq E,\qquad F_{r}\cup G_{j}^{\bullet}=E$

The former cannot have size $|E|-1$ because it is the union of a flat and a
coflat. Therefore $(F_{r}\cup G_{j}^{\bullet})-(F_{r}\cup
G_{j-1}^{\bullet})=(E-F_{r})\cap(G_{j}^{\bullet}-G_{j-1}^{\bullet})$ has size
at least $2$. But $\mathcal{F}$ and $\mathcal{G}$ are arboreal so
$\pi_{r+1}\cap\tau_{j}=(E-F_{r})\cap(G_{j}-G_{j-1})$ has size at most $1$.
This has two consequences:

a) $G_{j}^{\bullet}=G_{j}\cup 0$ and $G_{j-1}^{\bullet}=G_{j-1}$, that is,
$j=m$.

b) $0\in E-F_{r}=\pi_{r+1}$.

Similarly, consider the unions of the coflat $G_{n-r-1}^{\bullet}$ with the
flats in $\mathcal{F}$; let $i$ be the unique index such that

$F_{i-1}\cup G_{n-r-1}^{\bullet}\neq E,\qquad F_{i}\cup
G_{n-r-1}^{\bullet}=E.$

An analogous argument shows that $(F_{i}-F_{i-1})\cap(E-G_{n-r-1}^{\bullet})$
has size at least 2, whereas
$\pi_{i}\cap\tau_{n-r}=(F_{i}-F_{i-1})\cap(E-0-G_{n-r-1})$ has size at most
$1$. This has three consequences:

c) $G_{n-r-1}^{\bullet}=G_{n-r-1}$, that is, $m=n-r$.

d) $0\in F_{i}-F_{i-1}$, which in light of b) implies that $i=r+1$.

e) $(F_{i}-F_{i-1})\cap(E-0-G_{n-r-1})=\pi_{r+1}\cap\tau_{n-r}=\\{e\\}$ for
some element $e\in E-0$. But $e\in\pi_{r+1}$ means that $x_{e}=0$ is minimum
among all $x_{i}$s for any $(0,\mathsf{x})\in\sigma_{\mathcal{F}}$, and
$e\in\tau_{n-r}$ means that $y_{e}$ is minimum among all $y_{i}$s for any
$\mathsf{y}\in\sigma_{\mathcal{G}}$. Since $\mathsf{w}=\mathsf{x}+\mathsf{y}$
for some such $\mathsf{x}$ and $\mathsf{y}$, $w_{e}=x_{e}+y_{e}$ is minimum
among all $w_{i}$s, and since $\mathsf{w}$ is rapidly increasing, $e=1$.

It follows that in the intersection tree
$\Gamma_{{\boldsymbol{\pi}},{\boldsymbol{\tau}}}$, the top right vertex
$\pi_{r+1}$ contains $0$ and $1$ by d) and e), the bottom right vertex
$\tau_{n-r}$ contains $1$ by e), and thus $1$ connects them.

2\. Next we claim that for any path in the tree
$\Gamma_{{\boldsymbol{\pi}},{\boldsymbol{\tau}}}$ that ends with the edge $1$,
the first edge has the largest label.555This implies that the edge labels
decrease along any such path, but we will not use this in the proof. Assume
contrariwise, and consider a containment-minimal path $P$ that does not
satisfy this property; its edges must have labels satisfying
$e<f>f_{2}>\cdots>f_{k}$ sequentially. If edge $e$ goes from $\pi(e)$ to
$\tau(e)$, Lemma 2.3 gives $x_{e}=w_{e}-w_{f}\pm\text{(terms smaller than
$w_{f}$)}\approx-w_{f}<0=x_{1}$, contradicting that
$(0,\mathsf{x})\in\sigma_{\mathcal{F}}$. If $e$ goes from $\tau(e)$ to
$\pi(e)$, we get $y_{e}=w_{e}-w_{f}\pm\text{(terms smaller than
$w_{f}$)}\approx-w_{f}<w_{1}=y_{1}$, contradicting that
$\mathsf{y}\in\sigma_{\mathcal{G}}$.

3\. Now define

$\displaystyle b_{i}\coloneqq\min(F_{i}-F_{i-1})$ for $\displaystyle
i=1,\dots,r+1,$ $\displaystyle c_{j}\coloneqq\min(G_{j}-G_{j-1})$ for
$\displaystyle j=1,\ldots,n-r.$

Then $B\coloneqq\\{b_{1},\ldots,b_{r+1}\\}$ and
$C\coloneqq\\{c_{1},\ldots,c_{n-r}\\}$ are bases of $M$ and $N$, and
$\mathcal{F}=\mathcal{F}_{M}(B)$ and $\mathcal{G}=\mathcal{F}_{N}(C)$. We
claim that $B$ is $\beta$-nbc and $C=B^{\perp}$.

To do so, we first notice that the path from vertex $\pi_{i}=F_{i}-F_{i-1}$
(resp. $\tau_{j}=G_{j}-G_{j-1}$) to edge $1$ must start with edge $b_{i}$
(resp. $c_{j}$): if it started with some larger edge $b^{\prime}\in
F_{i}-F_{i-1}$, then the path from edge $b_{i}$ to edge $1$ would not start
with the largest edge. This has two consequences:

f) The sets $B$ and $C$ are disjoint. If we had $b_{i}=c_{j}=e$, then edge
$e$, which connects vertices $\pi_{i}=F_{i}-F_{i-1}$ and
$\tau_{j}=G_{j}-G_{j-1}$, would have to be the first edge in the paths from
both of these vertices to edge $1$; this is impossible in a tree. We conclude
that $B$ and $C$ are disjoint. Since $|B|=r+1$ and $|C|=n-r$, we have
$C=B^{\perp}$.

g) For each $i$ we have $x_{b_{i}}\approx w_{b_{i}}$, because the path from
$\tau_{i}$ to vertex $0$ – which is the path from $\tau_{i}$ to edge $1$, with
edge $1$ possibly removed – starts with the largest edge $b_{i}$, so Lemma 2.3
gives $x_{b_{i}}=w_{b_{i}}\pm\text{(smaller terms)}\approx w_{b_{i}}$.
Similarly $y_{c_{i}}\approx w_{c_{i}}$. Now,
$(0,\mathsf{x})\in\sigma_{\mathcal{F}}$ gives $x_{b_{1}}>\cdots>x_{b_{r+1}}$,
which implies $w_{b_{1}}>\cdots>w_{b_{r+1}}$, which in turn gives

$b_{1}>\cdots>b_{r}>b_{r+1};\qquad\textrm{ and analogously, }\qquad
c_{1}>\cdots>c_{n-r-1}>c_{n-r}=1.$

The former implies that $B$ is nbc in $M$ by 2.12. The latter, combined with
c), implies that $c_{1}>\cdots>c_{n-r-1}>0$ respectively are the minimum
elements of the flats
$G^{\bullet}_{1},\ldots,G^{\bullet}_{n-r-1},G^{\bullet}_{n-r}=E$ that they
sequentially generate, so $C\cup 0\setminus 1=B^{\perp}\cup 0\setminus 1$ is
nbc in $M^{\perp}$. It follows that $B$ is $\beta$-nbc in $M$.

We conclude that $B$ is $\beta$-nbc in $M$, $\mathcal{F}=\mathcal{F}_{M}(B)$,
and $\mathcal{G}=\mathcal{F}_{N}(B^{\perp})$, as desired. ∎

###### Proof of 1.2.1.

This follows by combining the previous two lemmas. ∎

##  4 Proof of the main theorem via torus-equivariant geometry

In this section we give a proof of 1.2.2 using the framework of _tautological
classes_ of matroids of Berget, Eur, Spink, and Tseng. See [BEST21] for
details on what follows. Recall that $M$ is a matroid on $E$ of rank $r+1$

In this framework, one works with the Chow ring of the permutohedral fan
$\Sigma_{E}$, which is the Bergman fan of the Boolean matroid on $E$. Its set
of maximal cones is in bijection with the set $\mathfrak{S}_{E}$ of
permutations of $E$. Let $S=\mathbb{Z}[t_{i}:i\in E]$; we can think of it as
the ring of polynomials on $\mathbb{R}^{E}$ with integer coefficients. Then
$S^{\mathfrak{S}_{E}}$ is the ring of $|E|!$-tuples of polynomials in $S$, one
polynomial $f_{\sigma}$ for each permutation $\sigma$ of $E$, or equivalently,
one polynomial $f_{\sigma}$ on each chamber $\sigma$ of $\Sigma_{E}$.

The _Chow ring_ $A^{\bullet}(\Sigma_{E})$ of $\Sigma_{E}$ has the following
description.

######  Definition 4.1.

Let $A^{\bullet}_{T}(\Sigma_{E})$ be the subring of $S^{\mathfrak{S}_{E}}$
defined by

$\displaystyle A_{T}^{\bullet}(\Sigma_{E})$ $\displaystyle=$
$\displaystyle\left\\{\text{continuous piecewise polynomials with integer
coefficients supported on }\Sigma_{E}\right\\}$ $\displaystyle=$
$\displaystyle\left\\{(f_{\sigma})_{\sigma\in\mathfrak{S}_{E}}\in
S^{\mathfrak{S}_{E}}\ \middle|\ \begin{matrix}\text{for any
$\sigma,\sigma^{\prime}\in\mathfrak{S}_{E}$, the polynomials $f_{\sigma}$ and
$f_{\sigma^{\prime}}$}\\\ \text{agree as functions on
$\sigma\cap\sigma^{\prime}\subseteq\mathbb{R}^{E}$}\end{matrix}\right\\}.$

Let $I$ be the ideal of $A_{T}^{\bullet}(\Sigma_{E})$ generated by the global
polynomials
$\\{(f_{\sigma})_{\sigma\in\mathfrak{S}_{E}}:f_{\sigma}=f_{\sigma^{\prime}}\text{
for all $\sigma\in\mathfrak{S}_{E}$}\\}$. Then

$A^{\bullet}(\Sigma_{E})=A_{T}^{\bullet}(\Sigma_{E})/I.$

One can associate to the fans $\Sigma_{(M,e)}$ and $-\Sigma_{(M/e)^{\perp}}$
certain elements $[\Sigma_{(M,e)}]$ and $[-\Sigma_{(M/e)^{\perp}}]$ of
$A^{\bullet}(\Sigma_{E})$ as follows. First, per 2.14, the fan $\Sigma_{E}$ in
$\mathbb{R}^{E}$ has lineality space $\mathbbm{1}\mathbb{R}$, and the quotient
fan $\Sigma_{E}/\mathbbm{1}\mathbb{R}$ is unimodularly isomorphic to the
_affine braid fan_ $\Sigma_{E,e}=\Sigma_{E}|_{x_{e}=0}$ in $\mathbb{R}^{E-e}$,
whose $|E|!$ chambers correspond to the possible orders of $\\{x_{f}\,:\,f\in
E-e\\}\cup\\{0\\}$. This is the affine Bergman fan of the Boolean matroid with
special element $e$.

Then, the fans $\Sigma_{(M,e)}$ and $-\Sigma_{(M/e)^{\perp}}$ are subfans of
$\Sigma_{E,e}$, and they are tropical fans in the sense that they satisfy the
balancing condition (see for instance [AHK18, Definition 5.1]). Via the theory
of Minkowski weights [FS97], they consequently define elements
$[\Sigma_{(M,e)}]$ and $[-\Sigma_{(M/e)^{\perp}}]$ of the Chow ring
$A^{\bullet}(\Sigma_{E})$. Moreover, the ring $A^{\bullet}(\Sigma_{E})$ is
equipped with a degree map
$\deg_{\Sigma_{E}}:A^{\bullet}(\Sigma_{E})\to\mathbb{Z}$, which agrees with
the map $\deg$ in 1.2 in the sense that

$\deg(\Sigma_{(M,e)}\cap-\Sigma_{(M/e)^{\perp}})=\deg_{\Sigma_{E}}([\Sigma_{(M,e)}]\cdot[-\Sigma_{(M/e)^{\perp}}]).$

For a survey of these facts, see [Huh18, Section 4], [AHK18, Section 5], or
[BEST21, Section 7.1].

We now describe how [BEST21] provided a distinguished representative in
$A_{T}^{\bullet}(\Sigma_{E})$ of the class $[\Sigma_{(M,e)}]\in
A^{\bullet}(\Sigma_{E})=A_{T}^{\bullet}(\Sigma_{E})/I$, and similarly for the
class $[-\Sigma_{(M/e)^{\perp}}]$. For a matroid $M$ on $E$, consider the
following elements of the rings $A_{T}^{\bullet}(\Sigma_{E})$ and
$A^{\bullet}(\Sigma_{E})$, modeled after the geometry of torus-equivariant
vector bundles from realizable matroids. For each permutation
$\sigma\in\mathfrak{S}_{E}$, let $B_{\sigma}(M)$ be the lexicographically
first basis of $M$ with respect to the ordering $\sigma(1)<\cdots<\sigma(n)$
of the ground set.

######  Definition 4.2.

[BEST21, Definition 3.9] Let $M$ be a matroid of rank $r+1$ on a ground set
$E$ of size $n+1$. Its _torus-equivariant tautological Chern classes_ are the
elements $\\{c_{i}^{T}(\mathcal{S}_{M}^{\vee})\\}_{i=0,\ldots,r+1}$ and
$\\{c_{j}^{T}(\mathcal{Q}_{M})\\}_{j=0,\ldots,n-r}$ in
$A_{T}^{\bullet}(\Sigma_{E})$ defined by

$\displaystyle c_{i}^{T}(\mathcal{S}_{M}^{\vee})_{\sigma}$
$\displaystyle=\text{the $i$-th elementary symmetric polynomial in
$\\{t_{k}:k\in B_{\sigma}(M)\\}$}\quad\text{and}$ $\displaystyle
c_{j}^{T}(\mathcal{Q}_{M})_{\sigma}$ $\displaystyle=\text{the $j$-th
elementary symmetric polynomial in $\\{-t_{\ell}:\ell\in E\setminus
B_{\sigma}(M)\\}$}$

for any permutation $\sigma\in\mathfrak{S}_{E}$. Their images in the quotient
$A^{\bullet}(\Sigma_{E})$, denoted $c_{i}(\mathcal{S}_{M}^{\vee})$ and
$c_{j}(\mathcal{Q}_{M})$, are called the _tautological Chern classes_ of $M$.

[BEST21, Proposition 3.8] shows that these elements are well-defined. The
results of [BEST21] yield the following representatives in
$A_{T}^{\bullet}(\Sigma_{E})$ of the elements $[\Sigma_{(M,e)}]$ and
$[-\Sigma_{(M/e)^{\perp}}]\in A^{\bullet}(\Sigma_{E})$. Let $M/e\oplus
U_{0,e}$ be the matroid on $E$ obtained from $M/e$ by adding back the element
$e$ as a loop. This matroid has rank $r$.

######  Lemma 4.3.

Let $M$ be a matroid of rank $r+1$ on a ground set $E$ of size $n+1$. Define
elements $[\Sigma_{(M,e)}]^{T}$ and $[-\Sigma_{(M/e)^{\perp}}]^{T}$ in
$A_{T}^{\bullet}(\Sigma_{E})$ by
$[\Sigma_{(M,e)}]^{T}=c_{n-r}^{T}(\mathcal{Q}_{M})$ and
$[-\Sigma_{(M/e)^{\perp}}]^{T}=c_{r}^{T}(\mathcal{S}^{\vee}_{M/e\,\oplus\,U_{0,e}})$,
or explicitly,

$[\Sigma_{(M,e)}]^{T}_{\sigma}=\prod_{i\in E\setminus
B_{\sigma}(M)}(-t_{i})\quad\text{and}\quad[-\Sigma_{(M/e)^{\perp}}]^{T}_{\sigma}={\prod_{i\in
B_{\sigma}(M/e\,\oplus\,U_{0,e})}}t_{i}\quad\text{for all
$\sigma\in\mathfrak{S}_{E}$}.$

Then, their images in the quotient $A^{\bullet}(\Sigma_{E})$ are exactly
$[\Sigma_{(M,e)}]$ and $[-\Sigma_{(M/e)^{\perp}}]$, respectively.

###### Proof.

The first equality is a restatement of [BEST21, Theorem 7.6] when one notes
that the choice of $e\in E$ induces an isomorphism
$\mathbb{R}^{E}/\mathbb{R}(1,\ldots,1)\simeq\mathbb{R}^{E-e}$. The second
statement also follows from that theorem when one combines it with [BEST21,
Propositions 5.11, 5.13], which describe how tautological Chern classes behave
with respect to matroid duality and direct sums, respectively. ∎

###### Proof of 1.2.2.

We begin with [BEST21, Theorem 6.2] which states that

$\deg_{\Sigma_{E}}\big{(}[\Sigma_{(M,e)}]\cdot
c_{r}(\mathcal{S}_{M}^{\vee})\big{)}=\beta(M).$

Thus, the desired statement
$\deg_{\Sigma_{E}}([\Sigma_{(M,e)}]\cdot[-\Sigma_{(M/e)^{\perp}}])=\beta(M)$
will follow once we show that
$[\Sigma_{(M,e)}]\cdot\big{(}c_{r}(\mathcal{S}_{M}^{\vee})-[-\Sigma_{(M/e)^{\perp}}]\big{)}$
= 0 in $A^{\bullet}(\Sigma_{E})$.

Towards this end, we look at the distinguished representative of this product
in $A_{T}^{\bullet}(\Sigma_{E})$, and show that the variable $t_{e}$ divides
$[\Sigma_{(M,e)}]_{\sigma}^{T}\cdot\big{(}c^{T}_{r}(\mathcal{S}_{M}^{\vee})_{\sigma}-[-\Sigma_{(M/e)^{\perp}}]^{T}_{\sigma}\big{)}$
for any $\sigma\in\mathfrak{S}_{E}$, as follows.

  * •

If $e\notin B_{\sigma}(M)$, then
$\displaystyle[\Sigma_{(M,e)}]^{T}_{\sigma}=\prod_{i\in E\setminus
B_{\sigma}(M)}(-t_{i})$ is divisible by $t_{e}$.

  * •

If $e\in B_{\sigma}(M)$, then $B_{\sigma}(M/e\oplus
U_{0,e})=B_{\sigma}(M)\setminus e$, and hence

$\displaystyle
c^{T}_{r}(\mathcal{S}_{M}^{\vee})_{\sigma}-[-\Sigma_{(M/e)^{\perp}}]^{T}_{\sigma}$
$\displaystyle=$ $\displaystyle\text{Elem}_{r}(\\{t_{k}\,:\,k\in
B_{\sigma}(M))-\prod_{j\in B_{\sigma}(M)\setminus e}t_{j}$ $\displaystyle=$
$\displaystyle\sum_{i\in B_{\sigma}(M)}\Big{(}\prod_{j\in
B_{\sigma}(M)\setminus i}t_{j}\Big{)}-\prod_{j\in B_{\sigma}(M)\setminus
e}t_{j}$ $\displaystyle=$ $\displaystyle\sum_{i\in B_{\sigma}(M)\setminus
e}\Big{(}\prod_{j\in B_{\sigma}(M)\setminus i}t_{j}\Big{)}$

is divisible by $t_{e}$.

This means that
$[\Sigma_{(M,e)}]^{T}\cdot\big{(}c^{T}_{r}(\mathcal{S}_{M}^{\vee})-[-\Sigma_{(M/e)^{\perp}}]^{T}\big{)}$
is a multiple of the global polynomial $t_{e}$, and hence is in the ideal $I$
of 4.1. Therefore
$[\Sigma_{(M,e)}]\cdot\big{(}c_{r}(\mathcal{S}_{M}^{\vee})-[-\Sigma_{(M/e)^{\perp}}]\big{)}=0$
in the quotient $A^{\bullet}(\Sigma_{E})$, as desired. ∎

##  5 Acknowledgments

We thank Bernd Sturmfels and Lauren Williams for organizing the Workshop on
Nonlinear Algebra and Combinatorics from Physics in April 2022 at the Harvard
University Center for the Mathematical Sciences and Applications. In
particular, Bernd asked the question that gave rise to this project.

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