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\usepackage{mathtools} %For coloneqq
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\newtheorem{thm}{Theorem}[section]
\newtheorem{prob}[thm]{Problem}
\newtheorem{defin}[thm]{Definition}
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\crefname{lm}{Lemma}{Lemmas}
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\newcommand{\w}{\mathsf{w}}
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\newcommand{\splamb}{\boldsymbol{\lambda}}
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%\usepackage{lipsum}
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\title[The tropical critical points of an affine matroid]{The tropical critical points of an affine matroid}
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%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details
\author[F. Ardila-Mantilla, C. Eur, and R. Penaguiao]{Federico Ardila-Mantilla\thanks{\href{mailto:federico@sfsu.edu}{federico@sfsu.edu}.
Partially supported by National Science Foundation Grant DMS-2154279.
}\addressmark{1}, Christopher Eur\thanks{\href{mailto:ceur@g.harvard.edu}{ceur@math.harvard.edu}. \hspace{-0.03cm}Partially \hspace{-0.03cm}supported \hspace{-0.03cm}by \hspace{-0.03cm}National \hspace{-0.03cm}Science \hspace{-0.03cm}Foundation Grant NSF DMS-2001854.}\addressmark{2}, \and Raul Penaguiao\thanks{\href{mailto:raul.penaguiao@mis}{raul.penaguiao@mis}. Partially supported by Schweizerischer Nationalfounds Grant P2ZHP2 191301.}\addressmark{3}}
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\address{\addressmark{1}Department of Mathematics, San Francisco State University and Universidad de Los Andes \\ \addressmark{2}Department of Mathematics, Harvard University \\ \addressmark{3}Max Planck Institute for Mathematics in the Sciences, Leipzig}
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\abstract{We prove that the maximum likelihood degree of a matroid $M$ equals its beta invariant $\beta(M)$. For an element $e$ of $M$ that is neither a loop nor a coloop, this is defined to be the degree of the intersection of the Bergman fan of $(M,e)$ and the inverted Bergman fan of $N=(M/e)^\perp$.
Equivalently, for a generic vector $\w \in \R^{E-e}$, 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.}
%
%\abstract{In this paper we study the generalization of a maximum likelihood degree of an abstract matroid.
%In the representable case, this corresponds to computing the number of critical points of a generic monomial function in the linear space corresponding to the matroid at hand, and was shown to be precisely the beta invariant of said matroid.
%
%Here, we present two proofs of this fact for general matroids, not only representable ones.
%In the first proof, we rephrase the problem as a counting problem of intersections between tropical fans.
%By using fan displacement and the combinatorial description the cones of the Bergman fan we relate this intersection number with certain bases of the matroid.
%In another proof, we compute the intersection point by recognizing that both sides of the equality are valuative functions.
%The desired follows by leveraging the fact that valuative functions are determined by representable matroids.}
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\keywords{matroid, Bergman fan, maximum likelihood degree}
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\begin{document}
\maketitle
\section{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, Sturmfels \cite{Sturmfels2022} posed one of those combinatorial problems that is deceivingly simple to state, but whose answer requires a deeper understanding of the objects at hand.
\begin{prob}\label{problem} \cite{Sturmfels2022} (Combinatorial version)
%Given a matroid $M$ on ground set $E$, and an element $e\in E$ that is neither a loop nor a coloop, and given a generic weight vector $\w \in \R^{E-e}$.
Let $M$ be a matroid on ground set $E$. Let $e$ 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.
Given a vector $\w \in \R^{E-e}$, find weight vectors
$(0,\x) \in \R^E$ on $M$ (where $e$ has weight $0$) and $\y \in \R^{E-e}$ on $N$ such that
$\bullet$ on each circuit of $M$, the minimum $\x$-weight occurs at least twice,
$\bullet$ on each circuit of $N$, the minimum $\y$-weight occurs at least twice, and
$\bullet$ $\w=\x+\y$.
\noindent Can this always be done? What is the number of solutions for generic $\w$?
%\begin{itemize}
%\item on each circuit of $M$, the minimum $\x$-weight occurs at least twice,
%\item on each circuit of $N$, the minimum $\y$-weight occurs at least twice, and
%\item $\w=\x+\y$.
%\end{itemize}
\end{prob}
%\begin{prob}\label{problem} \cite{Sturmfels2022}
%Let $G$ be a planar graph with edge set $E$, and pick out an edge $e\in E$ that is not a loop or a bridge.
%Let $G/e$ be the result of contracting edge $e$ in $G$, and let $H=(G/e)^\perp$ be the dual graph.
%Given a generic weight vector $\w \in \R^{E-e}$, we wish to find weights $(0,\x) \in \R^E$ on the edges of $G$ (giving edge $e$ weight $0$) and weights $\y \in \R^{E-e}$ on the edges of $H$ in such a way that:
%\begin{itemize}
%\item on each circuit of $G$, the minimum $\x$-weight occurs at least twice,
%\item on each circuit of $H$, the minimum $\y$-weight occurs at least twice, and
%\item $\w=\x+\y$.
%\end{itemize}
%\end{prob}
%This problem is one of those deceivingly simple combinatorial questions that is quite easy to state, but whose answer requires a deeper understanding of the objects at hand.
\begin{thm} \label{thm:main} (Geometric Version)
\Cref{problem} can always be solved. If $\w \in \R^{E-e}$ is generic, the number of solutions equals the \emph{beta invariant} $\beta(M)$ of the matroid.
\end{thm}
\begin{figure}[h]
\begin{center}
\includegraphics[scale=.9]{./FPSAC_figures/graph_matroid}
\qquad
\includegraphics[scale=.9]{./FPSAC_figures/graph_matroid_cntr}
\qquad
\includegraphics[scale=.8]{./FPSAC_figures/graph_matroid_cntrdual}
\end{center}
\caption{A graph $G$, its contraction $G/0$, and its dual $H=(G/0)^\perp$.\label{../img:dualmatroid}}
\end{figure}
We now restate \cref{thm:main} in tropical terms; see relevant definitions in \cref{sec:tropgeom}.
\begin{thm}\label{thm:dual}
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 affine matroid of $M$ with respect to $e$, and let $N = (M / e)^{\perp}$. Then the degree of the intersection of the Bergman fan $\Sigma_{(M,e)}$ and the inverted Bergman fan $-\Sigma_N$ is
%Then we compute the degree of the following intersection of tropical varieties in $\R^{E\setminus \{e\}}/_{\R \1}$:
\[
\deg( \Sigma_{(M,e)} \cdot -\Sigma_N) = \beta(M)\, .
\]
\end{thm}
Agostini, Brysiewicz, Fevola, K\"uhne, Sturmfels, and Telen \cite{agostini2021likelihood} first encountered (a special case of) \cref{problem} in their study of the maximum likelihood estimation for linear discrete models. Using algebro-geometric results of Huh and Sturmfels \cite{HuhSturmfels}, which built on work of Varchenko \cite{Varchenko}, they proved \cref{thm:main} and \cref{thm:dual} for matroids realizable over the real numbers.
We prove the equivalent \cref{thm:main} and \cref{thm:dual} for all matroids.
Following the original motivation, we call the answer to \cref{problem} the maximum likelihood degree of a matroid; our main result is that it equals the beta invariant.
We first prove \cref{thm:main} combinatorially, relying on the tropical geometric fact that all generic $\w$ give the same intersection degree. We show that when the entries of $\w$ are super-increasing with respect to some order $<$ on $E$, the solutions to \cref{problem} are naturally in bijection with the $\beta$-nbc bases of the matroid with respect to $<$.
We then sketch a proof of \Cref{thm:dual} that relies on the theory of tautological classes of matroids of Berget, Eur, Spink, and Tseng \cite{BEST}.
This is an extended abstract of our results in \cite{ardila2022maximumlikelihood}.
%
%\subsection{Geometric Motivation}
%
%\todo[inline]{Chris: I think you've written about maximum likelihood degrees, etc. Do you think you could write this section? }
%
%
%
%We write $\1$ for the all-one vector.
%We write $\Pi_e$ for the projection map resulting by dropping the $e$-th coordinate.
%Given a matroid, we consider $\Sigma_M$ its Bergman fan, defined in \cref{defin:bergfan}.
%\begin{thm}\label{thm:dual}
%Let $M$ be a matroid in $E$ of rank $r+1$, and consider $e\in E$ an element that is not a loop nor a coloop.
%Consider $N = (M / e)^{\perp}$.
%Then we compute the degree of the following intersection of tropical varieties in $\R^{E\setminus \{e\}}/_{\R \1}$:
%$$\deg( \Pi_e(\Sigma_M ) \cap -\Sigma_N) = \beta(M)\, .$$
%\end{thm}
%\raul{Should we use $|_{x_0 = 0}$ instead of the projection?}
%The structure of the paper is as follows.
%In \cref{sec:2} we introduce some preliminaries on set partitions and matroids. We introduce the intersection graph of two set partitions, which helps us understand how flats of the braid arrangement intersect.
%%We further lay down some important lemmas.
%%These account mainly for classical results in matroid theory and general linear algebra facts.
%In \cref{sec:3} we present the combinatorial proof of \cref{thm:main}, using the combinatorics of intersection graphs. In \cref{sec:4} we sketch the geometric proof of
%\cref{thm:dual} using the theory of matroid valuations and tautological classes of matroids.
\section{Notation and preliminaries\label{sec:2}}
\subsection{The lattice of set partitions}
A set partition $\splamb$ of a set $E$ is a collection of subsets, called blocks, of $E$, say $\splamb = \{\lambda_1 , \dots , \lambda_{\ell}\}$, that cover $E$ and the pairwise intersection is empty.
We write $\splamb \models E$. We let $|\splamb| = \ell$ be the number of blocks of $\splamb$.
If $e\in E$ and $\splamb \models E$, we write $\splamb(e)$ for the block of $\splamb $ that contains $e$.
We define the \emph{linear space of a set partition $\splamb = \{\lambda_1, \ldots, \lambda_\ell\} \models E$} to be
\begin{eqnarray*}
\L (\splamb ) &\coloneqq& \textrm{span}\{{\e}_{\lambda_1}, \ldots, {\e}_{\lambda_\ell}\} \, \subseteq \R^E \\
&=&
\{ \x \in \R^E \, \, | \, x_i = x_j \text{ whenever } i, j \text{ are in the same block of } \splamb\},
\end{eqnarray*}
where $\{\e_i \, : \, i \in E\}$ is the standard basis of $\R^E$, $\e_S = \sum_{s \in S} e_s$ for $S \subseteq E$.
Notice that $\dim \L (\splamb) = |\splamb|$.
The map $\splamb \mapsto \L(\splamb)$ is a bijection between the set partitions of $E$ and the flats of the % (reduced)
\emph{braid arrangement}, which is the hyperplane arrangement in $\R^E$
given by the hyperplanes $x_i=x_j$ for $i \neq j$ in $E$.
If $\splamb \models 0 \sqcup E$ then we write
$\L(\splamb)|_{x_0=0} = \{\x \in \R^E \, : \, (0, \x) \in \L(\splamb) \subseteq \R^{0 \sqcup E}\}$.
\subsection{The intersection graph of two set partitions}
The following construction from \cite{ArdilaEscobar} will play an important role.
\begin{defin}
Let $\splamb \models 0 \sqcup E$ and $\spmu \models E$ be set partitions.
The \emph{intersection graph} $\Gamma = \Gamma_{\splamb,\spmu}$
is the bipartite graph
with vertex set $\splamb \sqcup \spmu$ and edge set $E$, where the edge $e$ connects the parts $\lambda(e)$ of $\splamb$ and $\mu(e)$ of $\spmu$ containing $e$.
On this graph, the vertex corresponding to $\splamb(0)$ is marked with a hollow point.
\end{defin}
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 $E$. This is illustrated in \cref{fig:intgraph}.
\begin{figure}[h]
\begin{center}
\includegraphics[scale=1]{./FPSAC_figures/arboreal3} \qquad \qquad \includegraphics[scale=1]{./FPSAC_figures/arboreal3_vlabels}
\end{center}
\caption{The intersection graph of $\{6,59,2,013478\}\models [0,9]$ and $\{9,8,7,46,3,125\}\models [9]$, omitting brackets for easier legibility. \textbf{Left:} The elements of $[0, 9]$ are labelling the edges. \textbf{Right:} the vertices are labelled by parts of the set partitions. \label{fig:intgraph}}
\end{figure}
\begin{lm}\label{lm:arboreal}
Let $\splamb \models 0 \sqcup E$ and $\spmu \models E$ be set partitions and $\Gamma_{\splamb, \spmu}$ be their intersection graph.
\begin{enumerate}
\item
If $\Gamma_{\splamb, \spmu}$ has a cycle, then $\L(\splamb)|_{x_0=0} \cap (\w - \L(\spmu)) = \emptyset$ for generic\footnote{This means that this property holds for all $\w$ outside of a set of measure $0$.} $\w \in \R^E$.
\item
If $\Gamma_{\splamb, \spmu}$ is disconnected, then $\L(\splamb)|_{x_0=0} \cap (\w - \L(\spmu))$ is not a point for any $\w \in \R^E$.%/_{\R\1}$.
\item
If $\Gamma_{\splamb, \spmu}$ is a tree, then $\L(\splamb)|_{x_0=0} \cap (\w - \L(\spmu))$ is a point for any $\w \in \R^E$.
\end{enumerate}
\end{lm}
\begin{proof}
Let $\x \in \L(\splamb) $ and $\y \in \L(\mu)$ such that $\x + \y = \w$.
Write $x_{\lambda(i)} \coloneqq x_i$ and $y_{\mu(j)} \coloneqq y_j$ for simplicity.
The subspace
$\L(\splamb)|_{x_0=0} \cap (\w - \L(\spmu))$
%$\LL(\splamb) \cap (\w - \LL(\spmu))$
is cut out by the equalities
\begin{eqnarray*}
x_{\lambda(i)} + y_{\mu(i)} &=& w_i \qquad \textrm{ for } i \in E, \\
x_{\lambda(0)} &=& 0.
\end{eqnarray*}
This system has $|E|+1$ equations and $|\splamb| + |\spmu|$ unknowns. The linear dependences among these equations are controlled by the cycles of the graph $\Gamma_{\splamb, \spmu}$. More precisely, the first $|E|$ linear functionals $\{x_{\lambda(i)} + y_{\mu(i)} \, : \, i \in E \}$ gives a realization of the graphical matroid of $\Gamma_{\splamb, \spmu}$. The last equation is clearly linearly independent from the others.
If $\Gamma_{\splamb, \spmu}$ 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 $\w$, this equation does not hold, so $\L(\splamb)|_{x_0=0} \cap (\w - \L(\spmu))= \emptyset$.
If $\Gamma_{\splamb, \spmu}$ is disconnected, let $A$ be the set of edges in a connected component not containing the vertex $\lambda(0)$. If $\x \in \L(\splamb)$ and $\y \in \L(\spmu)$ satisfy $\x+\y= \w$ and $x_0=0$, then $\x+r\e_A \in \L(\splamb)$ and $\y-r\e_A \in \L(\spmu)$ also satisfy those equations for any real number $r$. Therefore $\L(\splamb)|_{x_0=0} \cap (\w - \L(\spmu))$ is not a point.
Finally, if $\Gamma_{\splamb, \spmu}$ is a tree, then its number of vertices is one more than the number of edges, that is, $|E| + 1 = |\splamb| + |\spmu|$, so the system of equations has equally many equations and unknowns. Also, these equations are linearly independent since $\Gamma_{\splamb, \spmu}$ is a tree. It follows that the system has a unique solution.
\end{proof}
When $\Gamma_{\splamb, \spmu}$ is a tree, we call $\splamb$ and $\spmu$ an \emph{arboreal pair}.
\begin{lm}\label{lm:xy}
Let $\splamb \models 0 \sqcup E$ and $\spmu \models E$ be an arboreal pair of set partitions and let $\Gamma_{\splamb, \spmu}$ be their intersection tree. Let $\w \in \R^E$. The unique vectors $\x \in \L(\splamb)$ and $\y \in \L(\spmu)$ such that $\x + \y = \w$ and $x_{0}=0$ are given by
\begin{eqnarray*}
x_{\lambda_i} &=& w_{e_1}-w_{e_2} + \cdots \pm w_{e_k} \qquad \textrm{ where $e_1e_2\ldots e_k$ is the unique path from $\lambda_i$ to $\lambda(0)$} \\
y_{\mu_j} &=& w_{f_1}-w_{f_2} + \cdots \pm w_{f_l} \qquad \textrm{ where $f_1f_2\ldots f_l$ is the unique path from $\mu_j$ to $\lambda(0)$}
\end{eqnarray*}
for any $i$ and $j$.
\end{lm}
\begin{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 $w_{e_i}$.
\end{proof}
\begin{smpl}
The set partitions $\splamb = \{6,59,2,013478\}\models [0,9]$, $\spmu = \{9,8,7,46,3,125\}\models~[9]$ form an arboreal pair, whose intersection tree is shown in \cref{fig:intgraph}. 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:
\begin{eqnarray*}
&&x_6 = w_6-w_4, \quad x_{59} = w_5-w_1, \quad x_2 = w_2-w_1, \quad x_{13478} = 0,\\
&&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.
\end{eqnarray*}
%where, where we double coordinates with the same value, for example, $x_{59} \coloneqq x_5=x_9$.
\end{smpl}
\begin{defin}\label{def:super}
A vector $\w \in \R^{n+1}$ is \emph{super-increasing} if $\omega_{i+1} > 3 \omega_i > 0$ for $1 \leq i \leq n$.
\end{defin}
\begin{lm}\label{lm:omega}
Let $\w$ be super-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
$
\omega_a + \sum_{i=1}^{a-1} \epsilon_i \omega_i <
\omega_b + \sum_{j=1}^{b-1} \delta_j \omega_j.
$
\end{lm}
\begin{proof} After verifying inductively that $\displaystyle \omega_c > 2 \sum_{i=1}^{c-1} \omega_i$ for all $c$, we see that
$$ \omega_a + \sum_{i=1}^{a-1} \epsilon_i \omega_i \leq \omega_a + \sum_{i=1}^{a-1} \omega_i < \frac32 \omega_a < \frac12 \omega_{b} < \omega_b - \sum_{j=1}^{b-1} \omega_j \leq \omega_b + \sum_{j=1}^{b-1} \delta_j \omega_j$$
as desired.
\end{proof}
\begin{defin}
Given a super-increasing vector $\w \in \R^{n+1}$ and a real number $x$, we will say $x$ \emph{is near} $w_i$ and write $x \approx w_i$ if $w_i - (w_1+\cdots + w_{i-1}) \leq x \leq w_i+(w_1+ \cdots + w_i)$. Note that if $x \approx w_i$ and $y \approx w_j$ for $i \dots > b_r\}$ is a basis of the matroid $M$.
We define the flats $F_i\coloneqq~\cl_M \{b_1, \dots, b_i\}$ and the flag $\FF_M(B) \coloneqq \{F_i\}$.
The following characterization of nbc-basis will be useful.
\begin{lm}\label{lm:nbc_flats}
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 $i = 1, \dots, r+1$.
\end{lm}
\begin{proof}
Omitted.
\end{proof}
An \emph{affine matroid} $(M,e)$ on $E$ is a matroid $M$ on $E$ with a chosen element $e \in E$.
\begin{defin}\cite{Sturmfels} \label{defin:bergfan}%[Bergman fan]
The \emph{Bergman fan} of a matroid $M$ on $E$ is
%We define the Bergman fan of $M$ as the following polyhedral fan
\[
\Sigma_M = \{\x \in \R^E \, | \, \min_{c\in C} x_c \text{ is attained at least twice for any circuit $C$ of $M$} \}\, .
\]
The \emph{Bergman fan of an affine matroid} $(M,e)$ on $E$ is
\[
\Sigma_{(M,e)} = \{\x \in \R^{E-e} \, | \, (0, \x) \in \Sigma_M\}.
\]
\end{defin}
The motivation for this definition comes from tropical geometry.
A subspace $V \subset \R^E$ determines a matroid $M_V$, and the tropicalization of $V$ is precisely the Bergman fan of $M_V$. Similarly, an affine subspace $W \subset \R^{E-e}$ determines an affine matroid $(M_W,e)$ consisting of a matroid $M_W$ on $E$ and a special element $e$, which represents the hyperplane at infinity. The tropicalization of $W$ is the Bergman fan $\Sigma_{(M_W,e)}$.
\begin{thm}\label{thm:bergman}\cite{ArdilaKlivans}
The Bergman fan of a matroid $M$ is a tropical fan equal to the union of the cones
\begin{eqnarray*}
\sigma_\FF &=& \textrm{cone}(\e_{F_1}, \ldots, \e_{F_{r+1}})\\
&=& \{ \x \in \R^E \, | \, x_a > x_b \text{ whenever $a\in F_i$ and $b\not\in F_i$ for some $1 \leq i \leq r+1$}\} \end{eqnarray*}
for the complete flags $\FF = \{\emptyset = F_0 \subsetneq F_1 \subsetneq \dots \subsetneq F_r \subsetneq F_{r+1} = E\}$ of flats of $M$.
\end{thm}
If $\Sigma_1$ and $\Sigma_2$ are tropical fans of complementary dimensions, then $\Sigma_1$ and $\Sigma_2+v$ intersect transversally at a finite set of points for generic vectors $v \in \R^n$. Each intersection point $p$ is equipped with a weight $w(p)$ that depends on the respective intersecting cones.
It is a general fact that the quantity
\[
\deg (\Sigma_1 \cdot \Sigma_2) := \sum_{p \in \Sigma_1 \cap (v+\Sigma_2)} w(p)
\]
is constant for generic $v$; this is the degree of the intersection \cite[Proposition 4.3.3, 4.3.6]{mikhalkin2010tropical}.
In all the intersections that arise in this paper, one can verify that the weight $w(p)$ is equal to 1 for every intersection point $p$. Therefore the degree of the intersection will be simply the number of intersection points:
\[
\deg (\Sigma_{(M,e)} \cdot - \Sigma_N) := | \Sigma_{(M,e)} \cap (v - \Sigma_N)|
\]
for generic $v \in \R^{E-e}$.
%\todo[inline]{We should say something short about tropical geometry, degrees, stable intersections, matroids, etc. In particular explain that $\Sigma_M$ is the tropicalization of a linear space is $\Sigma_M$ and the tropicalization of an affine linear space is $\Sigma_M|_{x_0=0}$.}
%
\section{A combinatorial proof of the main theorem \label{sec:3}}
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 $0$.
The following Lemma constructs an intersection point in $\Sigma_M|_{x_0=0} \cap (\w - \Sigma_N)$ for each $\beta$-nbc basis $B$ of $M$.
\begin{lm}\label{lm:bnbc_intersect}
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 $\w \in \R^{n}$ be super-increasing.
For any $\beta$-nbc basis $B$ of $M$,
there exist
unique vectors $(0,\x) \in \sigma_{\FF_M(B)}$ and $\y \in \sigma_{\FF_N(B^\perp)}$ such that $\x + \y = \w$.
\end{lm}
\begin{proof}
First we show that the set partitions $\opi$ of $\FF_M(B)$ and $\opi^\perp$ of $\FF_N(B^\perp)$ 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 $\Gamma_{\opi, \opi^\perp}$ is connected.
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 $\opi(a)$ in $\opi$, 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 \cref{lm:arboreal} that there exist unique $(0,\x) \in \L(\sppi)$ and $\y \in \L(\sppi^\perp)$ such that $\x + \y = \w$. It remains to show that $(0,\x) \in \sigma_\FF$ and $\y \in \sigma_{\FF^\perp}$.
\cref{lm:xy} provides formulas for $\x$ and $\y$ in terms of the paths from the various vertices of the tree of $\Gamma_{\sppi,\sppi^\perp}$ to $\pi(0)$. To understand those paths, let us give each edge $e$ an orientation as follows:
\begin{eqnarray*}
\pi(e) \longrightarrow \pi^\perp(e) & \text{ if } & \min \pi(e) > \min \pi^\perp(e), \\
\pi(e) \longleftarrow \pi^\perp(e) & \text{ if } & \min \pi(e) < \min \pi^\perp(e).
\end{eqnarray*}
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 $\opi$; 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 $\opi^{\perp}$.
Now, 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 $\pi(0)$ whose 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 \cref{lm:xy}, the first term dominates, and $x_b \approx w_b$. Similarly, $y_c \approx w_c$ for all $c \in B^\perp$.
Therefore, if we write $B = \{b_1 > \dots > b_r > b_{r+1}=0\}$, since $w$ is super-increasing, it follows that $x_{b_1} > x_{b_2} > \cdots > x_{b_r} > x_{b_{r+1}} = 0$, so indeed $(0,\x) \in \sigma_\FF$.
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 $\y \in \sigma_{\FF^\perp}$. The desired result follows.
\end{proof}
\begin{smpl}
The graphical matroid $M$ of the graph $G$ in \cref{../img:dualmatroid} has six $\beta$-nbc bases: 0256, 0257, 0259, 0368, 0378, and 0379. Let us compute the intersection point in $\Sigma_{(M,0)} \cap (\w - \Sigma_N)$ associated to $0257$ for the super-increasing vector $\w=(10^0, 10^1, \ldots, 10^8) \in \R^9$.
For $B = 0257$, we have $B^\perp = 134689$. Then
\begin{eqnarray*}
\FF_M(B) &=& \{\emptyset \subsetneq 7 \subsetneq 57 \subsetneq 2457 \subsetneq 0123456789 \} \\
\FF_{N}(B^\perp) &=& \{\emptyset \subsetneq 9\subsetneq 89 \subsetneq 689 \subsetneq 46789 \subsetneq 346789 \subsetneq 123456789 \},
\end{eqnarray*}
give rise to the corresponding set compositions
\[
\opi = 7|5|24|013689, \qquad \opi^\perp = 9|8|6|47|3|125.
\]
This is indeed an arboreal pair, as evidenced by their intersection graph in \cref{../img:arboreal1}.
\begin{figure}[h]
\begin{center}
\includegraphics[scale=1]{./FPSAC_figures/arboreal4}
\end{center}
\caption{The intersection graph of $\opi = 7|5|24|13689$ and $\opi^\perp = 9|8|6|47|3|125$.\label{../img:arboreal1}}
\end{figure}
\cref{lm:bnbc_intersect} gives us the unique points $(0,\x) \in \FF_{\opi}$ and $\y \in \FF_{\otau}$ such that $\x + \y = \w$; they are given by the paths to the special vertex $\pi(0)$ in the intersection tree $\Gamma_{\opi, \opi^\perp}$. 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{equation*}
\begin{array}{rrrrrrrrrr}
\x =& 0& 9 & 0 & 9 & 9999 & 0 & 999009 & 0 & 0 \\
\y =& 1& 1& 100& 991& 1& 100000& 991& 10000000& 100000000 \\
\w =& 1& 10& 100& 1000& 10000& 100000& 1000000& 10000000& 100000000
\end{array}
\end{equation*}
and $\x$ is in the intersection $\Sigma_{(M,0)} \cap (\w - \Sigma_N)$.
\end{smpl}
The following lemma indicates that any intersection point between $\Sigma_{(E,e)}$ and $v - \Sigma_N$ is an intersection between the cones described above.
That is, cones corresponding to a $\beta$-nbc basis.
\begin{lm}\label{lm:intersections_are_bnbc}
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 $\w \in \R^n$ be generic and super-increasing. Let
\begin{eqnarray*}
\FF &=& \{\emptyset = F_0 \subsetneq F_1 \subsetneq \dots \subsetneq F_r \subsetneq F_{r+1} = E \} \\
\GG &=& \{\emptyset = G_0 \subsetneq G_1 \subsetneq \dots \subsetneq G_{n-r-1} \subsetneq G_{n -r} = E - 0 \}
\end{eqnarray*}
be complete flags of the matroids $M$ and $N$, respectively, such that
$\Sigma_{(M,0)}$ and $\w - \Sigma_N$ intersect at $\sigma_{\FF}$ and $\w - \sigma_{\GG}$. Then there exists a $\beta$-nbc basis $B$ such that $\FF = \FF_M(B)$ and $\mathcal G = \FF_N(B^\perp)$.
\end{lm}
\begin{proof}
By \cref{lm:arboreal}, the set compositions $\opi$ and $\otau$ of $\FF$ and $\GG$ 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.
\medskip
1. Our first step will be to show that in the intersection tree $\Gamma_{\opi, \otau}$, 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 $G_i^{\bullet} \coloneqq \cl_{M^{\perp}}(G_i) \in \{G_i, G_i \cup 0\}$ is a flat of $M^\perp$. Consider the flag of flats of $M^\perp$
\[
\GG^{\bullet} \coloneqq \{ \emptyset = G_0^{\bullet} \subsetneq G_1^{\bullet} \subsetneq \dots \subsetneq G_{n-r-1}^{\bullet} \subsetneq G_{n-r}^{\bullet} = E\},
\]
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
\[
\GG^{\bullet} \coloneqq \{ \emptyset = G_0 \subsetneq G_1 \subsetneq \dots \subsetneq G_{M-1} \subsetneq G_M \cup 0 \subsetneq \dots \subsetneq G_{n-r-1} \cup 0 \subsetneq G_{n-r} \cup 0 = E\},
\]
Consider the unions of the flat $F_r$ with the coflats in $\GG^\bullet$; let $j$ be the index such that
\[
F_r \cup G_{j-1}^{\bullet} \neq E, \qquad F_r \cup G_j^{\bullet} = E
\]
Since it is the union of a flat and a coflat, the former cannot have size $|E|-1$, so $(F_r \cup G_j^{\bullet}) - (F_r \cup G_{j-1}^{\bullet}) = (E-F_r) \cap (G_j^{\bullet} - G_{j-1}^{\bullet})$
has size at least $2$. But $\FF$ and $\GG$ 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:
\smallskip
a) $G_j^{\bullet} = G_j \cup 0$ and $G_{j-1}^{\bullet} = G_{j-1}$, that is, $j=M$.
\smallskip
b) $0 \in E-F_r = \pi_{r+1}$.
\smallskip
Similarly, consider the unions of the coflat $G_{n-r-1}^{\bullet}$ with the flats in $\FF$; let $i$ be the index such that
\[
F_{i-1} \cup G_{n-r-1}^{\bullet} \neq E, \qquad F_i \cup G_{n-r-1}^{\bullet} = E.
\]
Analogously, we get 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:
\smallskip
c) $G_{n-r-1}^\bullet = G_{n-r-1} $, that is, $M=n-r$.
\smallskip
d) $0 \in F_i - F_{i-1}$, which in light of b) implies that $i=r+1$.
\smallskip
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 $(0,\x) \in \sigma_\FF$, and $e \in \tau_{n-r}$ means that $y_e$ is minimum among all $y_i$s for any $\y \in \sigma_\GG$. Since $\w=\x+\y$ for some such $\x$ and $\y$, $w_e=x_e+y_e$ is minimum among all $w_i$s, and since $\w$ is super-increasing, $e=1$.
\smallskip
It follows that in the intersection tree $\Gamma_{\opi, \otau}$, 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.
\medskip
2.
Next we claim that for any path in the tree $\Gamma_{\opi, \otau}$ that ends with the edge $1$, the first edge has the largest label.\footnote{It follows 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 be labelled $ef_2 > \cdots > f_k$ sequentially.
If edge $e$ goes from $\pi(e)$ to $\tau(e)$, \cref{lm:xy} gives $x_e = w_e-w_f \pm \text{(terms smaller than $w_f$)} \approx -w_f < 0 = x_1$, contradicting that $(0,\x) \in \sigma_\FF$. If $e$ goes from bottom to top, we get $y_e = w_e-w_f \pm \text{(terms smaller than $w_f$)} \approx -w_f < w_1 = y_1$, contradicting that $\y \in \sigma_\GG$.
\medskip
3. Now define
\begin{eqnarray*}
b_i \coloneqq \min (F_i - F_{i-1}) &\text{ for }& i=1, \dots, r+1, \\
c_j \coloneqq \min (G_j - G_{j-1}) &\text{ for }& j = 1, \ldots, n-r.
\end{eqnarray*}
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 $\FF = \FF_M(B)$ and $\GG = \FF_N(C)$. We claim that $B$ is $\beta$-nbc and $C=B^\perp$.
\smallskip
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 $b' \in F_i - F_{i-1}$, then the path from edge $b_i$ to edge $1$ would not start with the largest edge. This has two consequences:
\smallskip
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 %$0 = b_{r+1} \in B$,
$|B|=r+1$ and $|C|=n-r$, we have $C = B^\perp$.
\smallskip
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 \cref{lm:xy} gives $x_{b_i} = w_{b_i} \pm \text{(smaller terms)} \approx w_{b_i}$. Similarly $y_{c_i} \approx w_{c_i}$. Now, $(0, \x) \in \sigma_\FF$ 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 \cref{lm:nbc_flats}. 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 $C\cup0 \setminus 1 = B^\perp \cup 0 \setminus 1$ is nbc in $M^\perp$. It follows that $B$ is $\beta$-nbc in $M$.
\smallskip
We conclude that $B$ is $\beta$-nbc in $M$, $\FF = \FF_M(B)$, and $\GG = \FF_N(B^\perp)$, as desired.
\end{proof}
%This completes the combinatorial proof of our main result.
\begin{proof}[Combinatorial proof of \cref{thm:main}]
This follows from the previous two lemmas.
\end{proof}
%\begin{thm}
%Let $M$ be a matroid on $\{0, 1, \ldots, n\}$ where $0$ is neither a loop nor a coloop, and let $N=(M/e)^\perp$. Let $\w \in \R^n$ be super-increasing. Then there are exactly $\beta(M)$ choices of weights $(0,\x) \in \R^{[0,n]}$ on $M$ and $\y \in \R^n$ on $N$ satisfying the condition in \cref{problem}.
%%\begin{itemize}
%%\item on each circuit of $M$, the minimum $\x$-weight occurs at least twice,
%%\item on each circuit of $N$, the minimum $\y$-weight occurs at least twice, and
%%\item $\w=\x+\y$.
%%\end{itemize}
%\end{thm}
%
%\todo[inline]{Rephrase geometrically.}
%\todo[inline]{I think the above argument may work for arbitrary matroid morphisms: for matroids $M$ and $N'=M'/0$ where $M'$ is any matroid such that every flat of $M^\perp$ is a flat of $M^\perp$. It seems the only place we use duality is to say that $F \cup G^\bullet$ cannot have size $|E|-1$, and that would also work in that setting. There may be a more general result that the number of stable intersections of $\Sigma_M$ and $-\Sigma_{N'}$ is the \emph{beta invariant of the matroid morphism}?
%
%\emph{Logarithmic concavity for morphisms of matroids} by Eur-Huh and the last section of \emph{The closure of a linear space in a product of lines} by Ardila-Boocher seem relevant. Maybe we could derive the result in the next section directly from this?}
\section{A proof via torus-equivariant geometry} \label{sec:4}
We sketch a proof of \Cref{thm:dual} using the framework of \emph{tautological classes} of matroids of Berget, Eur, Spink, and Tseng. See \cite{ardila2022maximumlikelihood} for details on what follows.
\medskip
In this framework, one works with the permutohedral fan $\Sigma_E$, which is the Bergman fan of the Boolean matroid on $E$. Toric geometry equips $\Sigma_E$ with two ``cohomology'' rings: The torus-equivariant Chow ring $A^\bullet_T(\Sigma_E)$, and the non-equivariant Chow ring $A^\bullet(\Sigma_E)$, which is a quotient of $A^\bullet_T(\Sigma_E)$ by an explicitly described ideal $I$.
%
%Its set of maximal cones is in bijection with the set $\mathfrak S_E$ of permutations of $E$.
%Let $S = \Z[t_i: i\in E]$; we can think of it as the ring of polynomials on $\R^E$ 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 \emph{Chow ring} $A^\bullet(\Sigma_E)$ of $\Sigma_E$ has the following description.
%
%\begin{defin}\label{defin:chow}
%Let $A^\bullet_T(\Sigma_E)$ be the subring of
%$S^{\mathfrak S_E}$
%%$\prod_{\sigma \in \mathfrak S_E} \Z[t_i : i\in E]$
%defined by
%\begin{eqnarray*}
%A_T^\bullet(\Sigma_E) &=& \left\{\text{continuous piecewise polynomials with integer coefficients supported on } \Sigma_E \right\} \\
% &=& \left\{ (f_\sigma)_{\sigma \in \mathfrak S_E} \in S^{\mathfrak S_E} \ \middle| \ \begin{matrix}\text{for any $\sigma, \sigma'\in \mathfrak S_E$, the polynomials $f_\sigma$ and $f_{\sigma'}$}\\ \text{agree as functions on $\sigma \cap \sigma' \subseteq \R^E$}\end{matrix}\right\}.
%\end{eqnarray*}
%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'} \text{ for all $\sigma \in \mathfrak S_E$}\}$.
%Then
%\[
%A^\bullet(\Sigma_E) = A_T^\bullet(\Sigma_E)/I.
%\]
%\end{defin}
\medskip
First, one defines certain elements $[\Sigma_{(M,e)}]$ and $[-\Sigma_{(M/e)^\perp}]$ of $A^\bullet(\Sigma_E)$ by using the fact that the fans $\Sigma_{(M,e)}$ and $-\Sigma_{(M/e)^\perp}$ are subfans of $\Sigma_E$ that satisfy a balancing condition in the sense of Minkowski weights \cite{FS97}.
These elements have the following property: The ring $A^\bullet(\Sigma_E)$ is equipped with a degree map $\deg_{\Sigma_E}: A^\bullet(\Sigma_E) \to \Z$, which agrees with the map $\deg$ in \Cref{thm:dual} 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 \cite[Section 4]{Huh18b} or \cite[Section 7.1]{BEST}.
\medskip
Second, one notes that \cite{BEST} provided a distinguished representative in $A_T^\bullet(\Sigma_E)$ of the class $[\Sigma_{(M,e)}] \in A^\bullet(\Sigma_E) = A^\bullet_T(\Sigma_E)/I$, and similarly for the class $[-\Sigma_{(M/e)^\perp}]$.
In more detail, for a matroid $M$ of rank $r+1$ on a ground set $E$ of size $n+1$, \cite[Definition 3.9]{BEST} defined \emph{tautological Chern classes} of $M$ as certain 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 the equivariant Chow ring $A_T^\bullet(\Sigma_E)$, and established the following properties.
\begin{lm}\label{lem:classes}\cite[Theorem 7.6, Propositions 5.11, 5.13]{BEST}
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.
\end{lm}
%
%\begin{proof}
%The first equality is a restatement of \cite[Theorem 7.6]{BEST} when one notes that the choice of $e\in E$ induces an isomorphism $\R^E/\R(1,\ldots, 1) \simeq \R^{E-e}$. The second statement also follows from that theorem when one combines it with \cite[Propositions 5.11, 5.13]{BEST}, which describe how tautological Chern classes behave with respect to matroid duality and direct sums, respectively.
%%under the negation map $x\mapsto -x$ on $\R^E$, and under direct sums
%%
%%,
%% which describes in terms of direct sums of matroids how tautological Chern classes behave under the coordinate projection maps $\R^E \to \R^{E'}$.
%\end{proof}
%\begin{rem}
%Let $\mathcal S_M$ and $\mathcal Q_M$ be the tautological classes of matroids as defined in \cite{BEST}, and let $c_i$ denote their Chern classes.
%\Cref{lem:classes} is the torus-equivariant restatement of \cite[Proposition 5.11 \& Theorem 7.6]{BEST}, which stated that
%\[
%[\Sigma_{(M,e)}] = c_{|E|-r}(\mathcal Q_M) \quad\text{and}\quad [-\Sigma_{(M/e)^\perp}] = c_{r-1}(\mathcal S_{M/e \oplus U_{0,e}}^\vee).
%\]
%\end{rem}
Finally, to prove \Cref{thm:dual}, one begins with \cite[Theorem 6.2]{BEST} 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 one shows that $[\Sigma_{(M,e)}] \cdot \big(c_{r}(\mathcal S_M^\vee) - [-\Sigma_{(M/e)^\perp}]\big)$ = 0 in $A^\bullet(\Sigma_E)$.
For this end, one considers the distinguished representative $[\Sigma_{(M,e)}]^T \cdot \big(c^T_{r}(\mathcal S_M^\vee)- [-\Sigma_{(M/e)^\perp}]^T \big)$ of this product in $A_T^\bullet(\Sigma_E)$. An explicit description of this representative straightforwardly displays that it belongs to the ideal $I$.
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\end{document}