# Three results related to the half-plane property of matroids

Mario Kummer  Technische Universität, Dresden, Germany  mario.kummer@tu-
dresden.de and  David Sawall  Universität Konstanz  david.sawall@uni-
konstanz.de

###### Abstract.

We settle three problems from the literature on stable and real zero
polynomials and their connection to matroid theory. We disprove the weak real
zero amalgamation conjecture by Schweighofer and the second author. We
disprove a conjecture by Brändén and D’León by finding a relaxation of a
matroid with the weak half-plane property that does not have the weak half-
plane property itself. Finally, we prove that every quaternionic unimodular
matroid has the half-plane property which was conjectured by Pendavingh and
van Zwam.

###### 2020 Mathematics Subject Classification:

Primary: 05B35; Secondary: 12D10

Mario Kummer has been supported by the Deutsche Forschungsgemeinschaft under
Grant No. 502861109.

##  1\. Introduction

A _real zero polynomial_ is a polynomial
$P\in{\mathbb{R}}[x_{1},\ldots,x_{n}]$ with $P(0)\neq 0$ which has only real
zeros when restricted to any real line through the origin. Real zero
polynomials play an important role in the theory of semidefinite programming,
see e.g. [Vin12]. In this context, the second author and Schweighofer made the
following conjecture:

######  Conjecture ([SS24, Conjecture 7.6]).

Assume that $F\in{\mathbb{R}}[x_{1},x_{2},y_{1},\ldots,y_{m}]$ and
$G\in{\mathbb{R}}[x_{1},x_{2},z_{1},\ldots,z_{n}]$ are real zero polynomials.
If

$F|_{y_{1}=\cdots=y_{m}=0}=G|_{z_{1}=\cdots=z_{n}=0},$

then there is a real zero polynomial
$H\in{\mathbb{R}}[x_{1},x_{2},y_{1},\ldots,y_{m},z_{1},\ldots,z_{n}]$ such
that

$F=H|_{z_{1}=\cdots=z_{n}=0}\textrm{ and }G=H|_{y_{1}=\cdots=y_{m}=0}.$

We provide a counterexample to this conjecture in Section 3. Using the
connection between stable polynomials, a concept closely related to real zero
polynomials, and the theory of matroids and polymatroids established in
[COSW04] and [Brä07], we associate a polymatroid to every real zero
polynomial. Then the key step is 3.3 which says that if two real zero
polynomials satisfy the conclusion the above conjecture, then the associated
polymatroids can be _amalgamated_ in the sense of 2.4. Our counterexample
comes from two polymatroids which cannot be amalgamated. These polymatroids
are obtained by specializing the matroids $F_{7}^{-4}$ and $F_{7}^{-5}$ — two
relaxations of the Fano matroid which have the _half-plane property_ , meaning
that their bases generating polynomials are stable.

More generally, a matroid has the _weak half-plane property_ if its set of
bases agrees with the support of a stable polynomial. The question of which
matroids have the (weak) half-plane property has been extensively studied. In
this context, Brändén and D’León offered the following conjecture:

######  Conjecture ([BGD10, Conjecture 4.2]).

Suppose that $M$ has the weak half-plane property. Then so does any relaxation
of $M$.

In Section 4 we present a counterexample to this conjecture. More precisely,
we show that a relaxation $P_{1}$ of the matroid $P_{8}$ does not have the
weak half-plane property. On the other hand, $P_{8}$ does have the weak half-
plane property because it is representable over ${\mathbb{R}}$. For proving
that $P_{1}$ does not have the weak half-plane property, we employ the
techniques developed in [BGD10] for narrowing down the space of possible
coefficients of a hypothetical stable polynomial with support $P_{1}$.

A key feature of the matroid $P_{1}$ that made us examine it for the weak
half-plane property is that it is not representable — even under more general
notions of representability as discussed in [PVZ13]. Matroids representable
over more general structures than fields still often tend to have the weak
half-plane property, see e.g. [AB18]. Our third result is of this flavor.
Namely, we prove in Section 5 a conjecture which was attributed by Pendavingh
and van Zwam to David G. Wagner:

######  Conjecture ([PVZ13, Conjecture 6.9]).

All quaternionic unimodular matroids have the half-plane property.

The notion of quaternionic unimodular matroids is a generalization of the
class of sixth root of unity matroids to the skew field of quaternions. For a
precise defintion see 5.3.

##  2\. Preliminaries

We denote by ${\mathbb{N}}$ and ${\mathbb{N}}_{0}$ the set of positive and
nonnegative integers, respectively. In this section we let $E$ always denote a
finite set. For $i\in E$ we denote by $\delta_{i}\in{\mathbb{R}}^{E}$ the
$i$th unit vector. For $x\in{\mathbb{R}}^{E}$ we write $|x|=\sum_{i\in
E}|x_{i}|$. We further let $[n]=\\{1,\ldots,n\\}$ and denote by
$\binom{[n]}{k}$ the set of all $k$-element subsets of $[n]$ for every
$k,n\in{\mathbb{N}}$. We recall the cryptomorphic definitions of M-convex sets
and polymatroids.

######  Definition 2.1.

A subset $J\subseteq{\mathbb{N}}_{0}^{E}$ is _M-convex_ if for every $i\in E$
and every $\alpha,\beta\in J$ such that $\alpha_{i}>\beta_{i}$, there is $j\in
E$ satisfying

$\alpha_{j}<\beta_{j}\textrm{ and }\alpha-\delta_{i}+\delta_{j}\in J.$

######  Definition 2.2.

A _polymatroid_ on $E$ is a function $r\colon 2^{E}\to{\mathbb{N}}_{0}$ such
that we have for all $S,T\subseteq E$:

  1. (i)

$r(\emptyset)=0$,

  2. (ii)

$r(S)\leq r(T)$ if $S\subseteq T$, and

  3. (iii)

$r(S\cup T)+r(S\cap T)\leq r(S)+r(T)$.

If $J\subseteq{\mathbb{N}}_{0}^{E}$ is an M-convex set, then we define the
function $r_{J}\colon 2^{E}\to{\mathbb{N}}_{0}$ by

$r_{J}(S)=\max\\{\sum_{i\in S}\alpha_{i}\mid\alpha\in J\\}.$

Conversely, if $r\colon 2^{E}\to{\mathbb{N}}_{0}$ is a polymatroid, then we
define the set $J_{r}\subseteq{\mathbb{N}}_{0}^{E}$ by

$J_{r}=\\{x\in{\mathbb{N}}_{0}^{E}\mid\sum_{i\in S}x_{i}\leq r(S)\textrm{ for
all }S\subseteq E\textrm{ and }\sum_{i\in E}x_{i}=r(E)\\}.$

These two constructions are inverse to each other and define bijections
between the set of M-convex sets in ${\mathbb{N}}_{0}^{E}$ and the set of
polymatroids on $E$, see for example [Mur03, §4.4].

######  Remark 2.3.

A polymatroid $r\colon 2^{E}\to{\mathbb{N}}_{0}$ is the rank function of a
matroid $M$ on $E$ if and only if $r(\\{i\\})\leq 1$ for all $i\in E$. In this
case we have

$J_{r}=\\{\sum_{i\in B}\delta_{i}\mid B\textrm{ is a basis of }M\\}.$

We apply the definitions made for polymatroids to matroids by considering
their rank functions as polymatroids.

###  2.1. Amalgamation of polymatroids

There has been considerable interest in the question whether two
(poly)matroids can be amalgamated in the following sense, see for example
[PT82, Csi21] and also [Oxl11, §11.4].

######  Definition 2.4.

Let $E_{1},E_{2}$ be two finite sets, let $E_{0}=E_{1}\cap E_{2}$ and
$E_{3}=E_{1}\cup E_{2}$. Let $r_{i}$ be a polymatroid on $E_{i}$ for all
$i=0,1,2,3$.

  1. (a)

We say that $r_{0}$ is the _restriction_ of $r_{1}$ to $E_{0}$ if
$r_{0}(S)=r_{1}(S)$ for all $S\subseteq E_{0}$ and denote this by
$r_{0}=r_{1}|_{E_{0}}$.

  2. (b)

We say that $r_{3}$ is an _amalgam_ of $r_{1}$ and $r_{2}$ if
$r_{1}=r_{3}|_{E_{1}}$ and $r_{2}=r_{3}|_{E_{2}}$.

Clearly, a necessary condition for an amalgam of $r_{1}$ and $r_{2}$ as in 2.4
to exist is that $r_{1}|_{E_{0}}=r_{2}|_{E_{0}}$. However, this condition is
not sufficient [Oxl11, Example 7.2.4].

######  Definition 2.5.

A polymatroid $r_{0}$ on $E$ is called _sticky_ if there exists an amalgam for
all polymatroids $r_{1}$ and $r_{2}$ on finite sets $E_{1}$ and $E_{2}$ with
$E=E_{1}\cap E_{2}$ and such that $r_{0}=r_{i}|_{E}$ for $i=1,2.$

In order to describe conditions for a polymatroid to be sticky, recall the
following definition, generalizing the usual notion for matroids.

######  Definition 2.6.

Let $r$ be a polymatroid on $E$.

  1. (a)

A subset $F\subseteq E$ is called a _flat_ of $r$ if $r(F^{\prime})>r(F)$ for
every proper superset $F^{\prime}$ of $F$.

  2. (b)

Two flats $F_{1},F_{2}$ are called a _modular pair_ if

$r(F_{1})+r(F_{2})=r(F_{1}\cup F_{2})+r(F_{1}\cap F_{2}).$

######  Theorem 2.7 ([Csi21]).

Let $r$ be a polymatroid on $E$. If every pair of flats is modular, then $r$
is sticky. The converse holds if $|E|\leq 5$.

######  Example 2.8.

Let $2\leq n\leq 5$ and $E=[n]$. One checks that

$J=\\{x\in{\mathbb{N}}_{0}^{n}\mid|x|=3\textrm{ and }\forall i\in E:x_{i}\leq
2\\}$

is M-convex. The polymatroid $r_{J}$ is not sticky because the pair
$\\{1\\},\\{2\\}$ of flats is not modular.

###  2.2. Stable polynomials

We briefly recall the different stability and real zero properties of
polynomials and their relation to each other. As a general reference we
recommend the survey [Pem12].

######  Definition 2.9.

Let $P\in{\mathbb{R}}[x_{i}\mid i\in E]$ be a polynomial.

  1. (a)

The _support_ of $P$ is the unique subset
$\operatorname{supp}(P)\subseteq{\mathbb{N}}_{0}^{E}$ such that we can write

$P=\sum_{\alpha\in\operatorname{supp}(h)}c_{\alpha}x^{\alpha}$

for some non-zero $c_{\alpha}\in{\mathbb{R}}$.

  2. (b)

The polynomial $P$ is called _stable_ if for all $z\in{\mathbb{C}}^{E}$ such
that $\operatorname{Im}(z_{i})>0$ for all $i\in E$ we have $P(z)\neq 0$.

  3. (c)

The polynomial $P$ is called a _real zero polynomial_ if for all
$v\in{\mathbb{R}}^{E}$ the univariate polynomial $P(t\cdot
v)\in{\mathbb{R}}[t]$ has only real zeros.

  4. (d)

Let $P$ be homogeneous and $e\in{\mathbb{R}}^{E}$. Then $P$ is called
_hyperbolic_ with respect to $e$ if for all $v\in{\mathbb{R}}^{E}$ the
univariate polynomial $P(t\cdot e+v)\in{\mathbb{R}}[t]$ has only real zeros.

Stability is preserved under taking partial derivatives in coordinate
directions, under setting some variables equal to each other and under scaling
variables by positive scalars. We summarize the well-known connection between
the above concepts.

######  Proposition 2.10 (see for example [Pem12, Proposition 5.3]).

Let $P\in{\mathbb{R}}[x_{i}\mid i\in E]$ be a homogeneous polynomial. The
following are equivalent:

  1. (i)

$P$ is stable.

  2. (ii)

$P$ is hyperbolic with respect to every point in ${\mathbb{R}}_{>0}^{E}$.

  3. (iii)

$P$ is hyperbolic with respect to every point in ${\mathbb{R}}_{>0}^{E}$ and
every $e\in{\mathbb{R}}_{\geq 0}^{E}$ with $P(e)\neq 0$.

######  Lemma 2.11.

Let $P\in{\mathbb{R}}[x_{i}\mid i\in E]$ be a homogeneous polynomial and $i\in
E$. The following are equivalent:

  1. (i)

$P$ is hyperbolic with respect to $\delta_{i}$.

  2. (ii)

$P|_{x_{i}=1}\in{\mathbb{R}}[x_{i}\mid i\in E\smallsetminus\\{i\\}]$ is a real
zero polynomial.

######  Definition 2.12.

Let $P\in{\mathbb{R}}[x_{i}\mid i\in E]$ be hyperbolic with respect to
$e\in{\mathbb{R}}^{E}$. The _hyperbolicity cone_ of $P$ at $e$ is defined as

$\Lambda(P,e)=\\{v\in{\mathbb{R}}^{E}\mid P(t\cdot
e+v)\in{\mathbb{R}}[t]\textrm{ has only nonnegative zeros}\\}.$

Hyperbolicity cones are convex cones [Går59]. 2.10 can be rephrased in terms
of hypberbolicity cones.

######  Proposition 2.13.

Let $P\in{\mathbb{R}}[x_{i}\mid i\in E]$ be a homogeneous polynomial and
$e\in{\mathbb{R}}_{>0}^{E}$. The following are equivalent:

  1. (i)

$P$ is stable.

  2. (ii)

$P$ is hyperbolic with respect to $e$ and $\delta_{i}\in\Lambda(P,e)$ for all
$i\in E$.

The connection of stable polynomials to M-convex sets is given by the
following.

######  Theorem 2.14 ([Brä07, Theorem 3.2]).

Let $P\in{\mathbb{R}}[x_{i}\mid i\in E]$ be a homogeneous stable polynomial.
Then $\operatorname{supp}(P)$ is M-convex.

######  Definition 2.15.

Let $J\subseteq{\mathbb{N}}_{0}^{E}$ be an M-convex set.

  1. (a)

The _generating polynomial_ of $J$ is

$h_{J}=\sum_{\alpha\in J}\frac{x^{\alpha}}{\alpha!}\in{\mathbb{R}}[x_{i}\mid
i\in E]$

where $\alpha!=\prod_{i\in E}\alpha_{i}!$.

  2. (b)

We say that $J$ has the _half-plane property_ if its generating polynomial is
stable.

  3. (c)

We say that $J$ has the _weak half-plane property_ if there exists a
homogeneous stable polynomial $P\in{\mathbb{R}}[x_{i}\mid i\in E]$ such that
$J=\operatorname{supp}(P)$.

  4. (d)

We say that a polymatroid $r$ on $E$ has the _(weak) half-plane property_ if
the associated M-convex set $J_{r}$ has the (weak) half-plane property.

In the remaining part, we observe that in certain nice situations, restricting
the polymatroid of a homogeneous stable polynomial corresponds to plugging in
zeros for some of the variables.

######  Definition 2.16.

Let $J\subseteq{\mathbb{N}}_{0}^{E}$ be M-convex and let $T\subseteq E$. We
say that $J$ is _nondegenerate with respect to $T$_ if there exists $\alpha\in
J$ such that for all $i\in E\smallsetminus T$ we have $\alpha_{i}=0$.

######  Lemma 2.17.

Let $J\subseteq{\mathbb{N}}_{0}^{E}$ be M-convex and nondegenerate with
respect to $T\subseteq E$. Then we have $r_{J}|_{T}=r_{J^{\prime}}$ where

$J^{\prime}=\\{(\alpha_{i})_{i\in T}\mid\alpha\in J\textrm{ such that }\forall
i\in E\smallsetminus T:\alpha_{i}=0\\}\subseteq{\mathbb{N}}_{0}^{T}.$

###### Proof.

Let $S\subseteq T$. Then

$r_{J}|_{T}(S)=r_{J}(S)=\max\\{\sum_{k\in S}\alpha_{k}\mid\alpha\in J\\}.$

Let $\alpha\in J$ be a point where this maximum is attained such that
$\sum_{k\in E\smallsetminus T}\alpha_{k}$ is minimal. If $\sum_{k\in
E\smallsetminus T}\alpha_{k}=0$, then $\alpha\in J^{\prime}$ and we have
$r_{J}|_{T}(S)=r_{J^{\prime}}(S)$. Thus assume for the sake of a contradiction
that $\alpha_{i}>0$ for some $i\in E\smallsetminus T$. Since $J$ is
nondegenerate with respect to $T\subseteq E$, there exists $\beta\in J$ such
that for all $k\in E\smallsetminus T$ we have $\beta_{k}=0$. Because $J$ is
M-convex, there exists $j\in T$ such that $\alpha_{j}<\beta_{j}$ and
$\gamma=\alpha-\delta_{i}+\delta_{j}\in J$. Then $\sum_{k\in
S}\gamma_{k}\geq\sum_{k\in S}\alpha_{k}$ because $i\not\in S$ and $\sum_{k\in
E\smallsetminus T}\gamma_{k}<\sum_{k\in E\smallsetminus T}\alpha_{k}$ because
$i\in E\smallsetminus T$ and $j\in T$. This contradicts our minimality
assumption. ∎

######  Corollary 2.18.

Let $E=E_{1}\cup E_{2}$ and let $P\in{\mathbb{R}}[x_{i}\mid i\in E]$ be a
polynomial such that $J=\operatorname{supp}(P)$ is M-convex. Finally, for
$k=1,2$, we let

$P_{k}=P|_{x_{i}=0\textrm{ for }i\not\in E_{k}}\in{\mathbb{R}}[x_{i}\mid i\in
E_{k}]$

and $J_{k}=\operatorname{supp}(P_{k})$. If $P_{1},P_{2}$ are both not
identically zero, then $J_{1}$ and $J_{2}$ are M-convex and $r_{J}$ is an
amalgam of $r_{J_{1}}$ and $r_{J_{2}}$.

###### Proof.

The condition that $P_{k}$ is not identically zero is equivalent to $J$ being
nondegenerate with respect to $E_{k}$. Now the claim follows from 2.17. ∎

##  3\. Real zero amalgamation

In this section we always let $E=E_{1}\cup E_{2}$ be a finite set and
$E_{0}=E_{1}\cap E_{2}$. We assume that $0\in E_{0}$. For $k=0,1,2$ we let
$0\neq P_{k}\in{\mathbb{R}}[x_{i}\mid i\in E_{k}]$ be homogeneous and stable
such that

$P_{0}=P_{k}|_{x_{i}=0\textrm{ for }i\not\in E_{0}}$

for $k=1,2$. Note that this implies that $P_{0},P_{1}$ and $P_{2}$ all have
the same degree $d$. We further denote $J_{k}=\operatorname{supp}(P_{k})$ and
$r_{k}=r_{J_{k}}$ for $k=0,1,2$. For $k=0,1,2$ we also consider the polynomial
$H_{k}$ obtained from $P_{k}$ by substituting $x_{0}+x_{i}$ for $x_{i}$ for
all $i\in E_{0}\smallsetminus\\{0\\}$.

######  Lemma 3.1.

The polynomial $H_{k}$ is hyperbolic with respect to $\delta_{0}$. Its
hyperbolicity cone contains $\delta_{i}$ for all $i\in E_{k}$ and the point

$\delta_{0}-\sum_{0\neq i\in E_{0}}\delta_{i}.$

###### Proof.

The statement of $H_{k}$ being hyperbolic with respect to $\delta_{0}$ is
equivalent to $P_{k}$ being hyperbolic with respect to $\sum_{i\in
E_{0}}\delta_{i}$. Because of

$P_{k}\left(\sum_{i\in E_{0}}\delta_{i}\right)=P_{0}\left(\sum_{i\in
E_{0}}\delta_{i}\right)\neq 0,$

by stability of $P_{0}$, this follows because $P_{k}$ is stable. The
hyperbolicity cone of $H_{k}$ containing $\delta_{i}$, $i\in E_{k}$, follows
from the same statement for $P_{k}$. The last statement is equivalent to
$\delta_{0}$ being in the hyperbolicity cone of $P_{k}$. ∎

######  Lemma 3.2.

Let $H\in{\mathbb{R}}[x_{i}\mid i\in E]$ be hyperbolic with respect to
$\delta_{0}$ such that the hyperbolicity cone of

$H|_{x_{i}=0\textrm{ for }i\not\in E_{k}}$

agrees with the hyperbolicity cone of $H_{k}$ for $k=1,2$. Then the
hyperbolicity cone of $H$ contains $\delta_{i}$ for all $i\in E$ and the point

$\delta_{0}-\sum_{0\neq i\in E_{0}}\delta_{i}.$

###### Proof.

This follows immediately from 3.1. ∎

Now let $E^{\prime}=E\smallsetminus\\{0\\}$ and likewise
$E_{k}^{\prime}=E_{k}\smallsetminus\\{0\\}$ for $k=0,1,2$. Then we consider
the polynomials

$Q_{k}=H_{k}|_{x_{0}=1}\in{\mathbb{R}}[x_{i}\mid i\in E_{k}^{\prime}]$

for $k=0,1,2$. These are real zero polynomials with the property that

$Q_{0}=Q_{k}|_{x_{i}=0\textrm{ for }i\not\in E^{\prime}_{0}}$

or $k=1,2$. We have the following result.

######  Theorem 3.3.

Assume that there is a real zero polynomial $Q\in{\mathbb{R}}[x_{i}\mid i\in
E^{\prime}]$ with

$Q_{k}=Q|_{x_{i}=0\textrm{ for }i\not\in E^{\prime}_{k}}$

for $k=1,2$. Then $r_{1}$ and $r_{2}$ have an amalgam.

###### Proof.

Let $d^{\prime}=\deg(Q)$. We define

$H=x_{0}^{d^{\prime}}\cdot Q(\frac{x_{i}}{x_{0}}\mid i\in
E^{\prime})\in{\mathbb{R}}[x_{i}\mid i\in E].$

Then we have

$H|_{x_{i}=0\textrm{ for }i\not\in E^{\prime}_{k}}=x_{0}^{d^{\prime}-d}\cdot
H_{k}$

for $k=1,2$. Thus by 3.2 the hyperbolicity cone of $H$ contains $\delta_{i}$,
$i\in E$, and the point

$\delta_{0}-\sum_{0\neq i\in E_{0}}\delta_{i}.$

This implies that the polynomial $P$ obtained from $H$ by substituting
$x_{i}-x_{0}$ for $x_{i}$ for all $i\in E_{0}\smallsetminus\\{0\\}$ is stable.
We have

$P|_{x_{i}=0\textrm{ for }i\not\in E_{k}}=x_{0}^{d^{\prime}-d}\cdot P_{k}$

for $k=1,2$. Finally, we let $P^{\prime}$ be the polynomial obtained from $P$
by dropping all monomials that are not divisible by $x_{0}^{d^{\prime}-d}$ and
dividing the result by $x_{0}^{d^{\prime}-d}$. Then

$P^{\prime}|_{x_{i}=0\textrm{ for }i\not\in E_{k}}=P_{k}$

for $k=1,2$. The support of $P^{\prime}$ is M-convex because it agrees with
the support of the stable polynomial $\frac{\partial^{d^{\prime}-d}}{\partial
x_{0}}P$. Now the claim follows from 2.18. ∎

Now we are ready to disprove the real zero amalgamation conjecture from
[SS24].

######  Conjecture 3.4 ([SS24, Conjecture 7.6]).

Let $E^{\prime}=E_{1}^{\prime}\cup E_{2}^{\prime}$ be a finite set such that
$E_{0}^{\prime}=E_{1}^{\prime}\cap E_{2}^{\prime}$ has two elements. For
$k=1,2$ let $Q_{k}\in{\mathbb{R}}[x_{i}\mid i\in E_{k}^{\prime}]$ be a real
zero polynomial. If

$Q_{1}|_{x_{i}=0\textrm{ for }i\not\in E_{0}^{\prime}}=Q_{2}|_{x_{i}=0\textrm{
for }i\not\in E_{0}^{\prime}},$

then there is a real zero polynomial $Q\in{\mathbb{R}}[x_{i}\mid i\in
E^{\prime}]$ such that

$Q_{k}=Q|_{x_{i}=0\textrm{ for }i\not\in E_{k}^{\prime}}$

for $k=1,2$.

3.3 shows that for $|E_{0}|=3$ (and hence $|E_{0}^{\prime}|=2$) every choice
of $P_{k}$, $k=0,1,2$, as above such that additionally $r_{1}$ and $r_{2}$ do
not have an amalgam gives a counterexample to 3.4.

310′2′021′

410′2′021′

Figure 1. The matroids $F_{7}^{-4}$ (left) and $F_{7}^{-5}$ (right).

###  3.1. An example with $|E_{0}|=3$

We consider the matroids $F_{7}^{-4}$ and $F_{7}^{-5}$ defined as in Figure 1.
These matroids both have the half-plane property, see [COSW04, §A.2.2] for
$F_{7}^{-5}$ and [WW09] for $F_{7}^{-4}$. This means that their generating
polynomials $h_{F_{7}^{-4}}$ and $h_{F_{7}^{-5}}$ are stable. We let $P_{1}$
and $P_{2}$ be the polynomials obtained from $h_{F_{7}^{-4}}$ and
$h_{F_{7}^{-5}}$, respectively, by setting $x_{0^{\prime}}=x_{0}$,
$x_{1^{\prime}}=x_{1}$ and $x_{2^{\prime}}=x_{2}$. By construction we have

$P_{1}|_{x_{3}=0}=P_{2}|_{x_{4}=0}.$

Hence $E_{1}=\\{0,1,2,3\\}$ and $E_{2}=\\{0,1,2,4\\}$. Note that the support
of $P_{0}:=P_{1}|_{x_{3}=0}$ is the M-convex set from 2.8 for $n=3$. We will
show that the polymatroids $r_{1}$ and $r_{2}$ corresponding to $P_{1}$ and
$P_{2}$ do not have an amalgam so that the real zero polynomials $Q_{1}$ and
$Q_{2}$ obtained from $P_{1}$ and $P_{2}$ constitute a counterexample to 3.4.
In fact, we will even prove that for every $m\in{\mathbb{N}}$ the polymatroids
$m\cdot r_{1}$ and $m\cdot r_{2}$ do not have an amalgam which shows that
$Q_{1}^{m}$ and $Q_{2}^{m}$ also do not satisfy the conclusion of 3.4 for any
$m\in{\mathbb{N}}$.

######  Theorem 3.5.

The polymatroids $m\cdot r_{1}$ and $m\cdot r_{2}$ do not have an amalgam.

###### Proof.

We proceed as in the proof of [PT82, Theorem 2]. Assume for the sake of a
contradiction that the polymatroid $r$ on $E=\\{0,1,2,3,4\\}$ is an amalgam of
$m\cdot r_{1}$ and $m\cdot r_{2}$. We have

$r(\\{0\\})+r(\\{0,3\\})\leq r(\\{0\\})+r(\\{0,3,4\\})\leq
r(\\{0,3\\})+r(\\{0,4\\}).$

By definition of $r_{1}$ and $r_{2}$ this implies that

$2m+2m\leq 2m+r(\\{0,3,4\\})\leq 2m+2m.$

Thus we have $r(\\{0,3,4\\})=2m$. Likewise one shows that $r(\\{2,3,4\\})=2m$.
Furthermore, we have that

$4m\leq r(\\{3,4\\})+3m\leq r(\\{3,4\\})+r(\\{0,2,3,4\\})\leq
r(\\{0,3,4\\})+r(\\{2,3,4\\})=4m.$

This shows $r(\\{3,4\\})=m$. Finally, we have

$m+2m\leq r(\\{3\\})+r(\\{1,3,4\\})\leq r(\\{1,3\\})+r(\\{3,4\\})=2m+m$

which implies $r(\\{1,3,4\\})=2m$ contradicting

$3m=r(\\{1,4\\})\leq r(\\{1,3,4\\}).$

Hence the polymatroids $m\cdot r_{1}$ and $m\cdot r_{2}$ do not have an
amalagam. ∎

##  4\. Weak half-plane property

Let $E$ always denote a finite set and $M$ a matroid on $E$ with set of bases
${\mathcal{B}}$.

######  Definition 4.1 ([BGD10]).

We say that four bases $B_{1},B_{2},B_{3},B_{4}\in{\mathcal{B}}$ form a
_degenerate quadrangle_ of $M$ if there exists $S\subseteq E$ and pairwise
different $i,j,k,l\notin S$ such that

$(B_{1},B_{2},B_{3},B_{4})=(S\cup\\{i,k\\},S\cup\\{j,l\\},S\cup\\{i,l\\},S\cup\\{j,k\\})$

and if at most one of $S\cup\\{i,j\\}$ and $S\cup\\{k,l\\}$ is a bases of $M$.

The following theorem was used in [BGD10] to reduce the number of possible
parameters when searching for a stable polynomial with support $M$.

######  Theorem 4.2 ([Brä07]).

For every basis $B\in{\mathcal{B}}$ let $0\neq a_{B}\in{\mathbb{R}}$ be such
that the multiaffine and homogeneous polynomial

$P=\sum_{B\in{\mathcal{B}}}a_{B}\cdot\prod_{i\in
B}x_{i}\in{\mathbb{R}}[x_{i}\mid i\in E]$

is stable. If $B_{1},B_{2},B_{3},B_{4}$ form a degenerate quadrangle of $M$,
then

(1) $a_{B_{1}}a_{B_{3}}=a_{B_{2}}a_{B_{4}}.$

Letting $b_{B}:=\log(|a_{B}|)$ for all $B\in{\mathcal{B}}$ we obtain from
Equation 1 linear equations

$b_{B_{1}}+b_{B_{3}}-b_{B_{2}}-b_{B_{4}}=0$

for all degenerate quadrangles $B_{1},B_{2},B_{3},B_{4}$ of $M$. We denote by
$V_{M}\subseteq{\mathbb{R}}^{{\mathcal{B}}}$ the linear space cut out by all
such equations. By [BGD10, Lemma 2.6] the vector space

$W_{M}:=\\{(\sum_{i\in B}v_{i})_{B\in{\mathcal{B}}}\mid
v\in{\mathbb{R}}^{E}\\}$

is an $(|E|-z+1)$-dimensional linear subspace of $V_{M}$ where $z$ is the
number of connected components of $M$.

######  Lemma 4.3.

Let $U_{M}$ be a linear complement of $W_{M}$ in $V_{M}$. If $M$ has the weak
half-plane property, then there exists a vector $b\in U_{M}$ such that

$\sum_{B\in{\mathcal{B}}}\exp(b_{B})\cdot\prod_{i\in B}x_{i}$

is stable.

###### Proof.

This follows in the same way as [BGD10, Theorem 2.3]: Scaling the variables by
$x_{i}\mapsto\exp(v_{i})x_{i}$ corresponds, after taking logarithms of the
coefficients, to shifting by the corresponding vector from $W_{M}$. ∎

For choosing a linear complement of $W_{M}$ in $V_{M}$ in a nice way, the
following lemma might be useful.

######  Lemma 4.4.

Let $M$ be represented by a matrix $A\in{\mathbb{R}}^{d\times|E|}$ of rank
$d$. For $B\in{\mathcal{B}}$ we denote by $A[B]$ the corresponding $d\times d$
submatrix. Then

$u(A):=(\log|\det(A[B])|)_{B\in{\mathcal{B}}}\in V_{M}.$

Furthermore, if $M$ is not regular, then $u(A)\not\in W_{M}$.

###### Proof.

By [COSW04, Theorem 8.1] the polynomial

$\sum_{B\in{\mathcal{B}}}\det(A[B])^{2}\cdot\prod_{i\in B}x_{i}$

is stable. This proves the first claim. Now assume that there exists
$v\in{\mathbb{R}}^{E}$ such that $u(A)=(\sum_{i\in
B}v_{i})_{B\in{\mathcal{B}}}$. Scaling the $i$th column of $A$ by
$\exp(-v_{i})$ for all $i\in E$, we obtain a matrix $A^{\prime}$ representing
$M$ all of whose maximal minors are in $\\{-1,0,1\\}$. After multiplication of
$A^{\prime}$ from the left by a suitable invertible matrix we can additionally
assume that $A^{\prime}[B_{0}]$ is the identity matrix for some
$B_{0}\in{\mathcal{B}}$. Then $A^{\prime}$ is a totally unimodular matrix
representing $M$ which shows that $M$ is regular. ∎

######  Example 4.5.

In this example, we consider the rank 4 matroid $M=P_{8}$ on 8 elements which
is represented by the real matrix

$A:=\begin{pmatrix}1&0&0&0&0&1&1&2\\\ 0&1&0&0&1&0&1&1\\\ 0&0&1&0&1&1&0&1\\\
0&0&0&1&2&1&1&0\end{pmatrix}$

whose columns we label by $0,\ldots,7$. This matroid is not regular [COSW04,
§A4] and therefore $u(A)$ is in $V_{M}$ but not in $W_{M}$ by 4.4. Using the
Macaulay2 [GS] package “Matroids” [Che18] we compute that $\dim(V_{M})=9$.
Because $M$ is connected, this implies that the span of $u(A)$ is a linear
complement of $W_{M}$ in $V_{M}$. Note that $P_{8}$ has the weak half-plane
property because it is representable over ${\mathbb{R}}$.

Recall that if $X$ is a circuit-hyperplane of $M$, then
${\mathcal{B}}\cup\\{X\\}$ is the set of bases of a matroid $M^{\prime}$
[Oxl11, Theorem 1.5.14]. Then $M^{\prime}$ is called a _relaxation_ of $M$.

######  Lemma 4.6.

Let $M^{\prime}$ be a relaxation of $M$. Then every degenerate quadrangle of
$M^{\prime}$ is a degenerate quadrangle of $M$.

###### Proof.

We proceed as in the proof of [D’L09, Lemma 3.43]. Denote by $X$ the circuit-
hyperplane of $M$ such that $M^{\prime}$ is the relaxation of $M$ by $X$. We
show that for all $x\in X$ and $y\in E\smallsetminus X$ the set

$(X\smallsetminus\\{x\\})\cup\\{y\\}$

forms a basis of $M$ (and therefore of $M^{\prime}$). Then it immediately
follows that $X$ cannot be contained in some degenerate quadrangle of
$M^{\prime}$. Since $X$ is a circuit, $X\smallsetminus\\{x\\}$ still has rank
$\operatorname{rank}(M)-1=\operatorname{rank}(M^{\prime})-1$. Then
$(X\smallsetminus\\{x\\})\cup\\{y\\}$ has rank
$\operatorname{rank}(M^{\prime})$ because $X$ is closed and $y\notin X$. Thus
$(X\smallsetminus\\{x\\})\cup\\{y\\}$ is a basis of $M^{\prime}$. ∎

######  Remark 4.7.

Assume that $\operatorname{rank}(M)\geq 2$. Let $M^{\prime}$ be the relaxation
of $M$ by the circuit-hyperplane $X$ and denote
${\mathcal{B}}^{\prime}={\mathcal{B}}\cup\\{X\\}$. 4.6 implies that the map
${\mathbb{R}}^{{\mathcal{B}}}\to{\mathbb{R}}^{{\mathcal{B}}^{\prime}}$, that
sends $v\in{\mathbb{R}}^{{\mathcal{B}}}$ to the vector $w$ with $w_{X}=0$ and
$w_{B}=v_{B}$ for all $B\in{\mathcal{B}}$, maps $V_{M}$ to $V_{M^{\prime}}$.
We thus obtain a natural embedding

$\iota\colon V_{M}\hookrightarrow V_{M^{\prime}}.$

Furthermore, we have $\delta_{X}\in V_{M^{\prime}}$. If $U_{M}$ is a linear
complement of $W_{M}$ in $V_{M}$, then

$\left(\iota(U_{M})\oplus{\mathbb{R}}\cdot\delta_{X}\right)\cap
W_{M^{\prime}}=\\{0\\}.$

However, in general $\iota(U_{M})$ and $\delta_{X}$ do not span a linear
complement of $W_{M^{\prime}}$, see for example [BGD10, Table 1].

######  Example 4.8.

As an illustration, we recall [BGD10, Example 4.1]. The non-Fano matroid
$M^{\prime}=F_{7}^{-}$ is represented by the real matrix

$A=\begin{pmatrix}1&1&0&0&0&1&1\\\ 0&1&1&1&0&0&1\\\
0&0&0&1&1&1&1\end{pmatrix}$

whose columns we label by $1,\ldots,7$. This matroid is not regular [COSW04,
§A.2.2] and therefore $u(A)$ is in $V_{M^{\prime}}$ but not in
$W_{M^{\prime}}$ by 4.4. On the other hand, the non-Fano matroid is a
relaxation of the Fano matroid $F_{7}$ by the circuit hyperplane
$\\{2,4,6\\}$. Because $\dim(V_{M^{\prime}}/W_{M^{\prime}})=1$ by [BGD10], it
follows from 4.7 that $u(A)+W_{M^{\prime}}$ must contain a scalar multiple of
$\delta_{\\{2,4,6\\}}$. Indeed, one has that
$u(A)=\log(2)\cdot\delta_{\\{2,4,6\\}}$. It was further noted in [BGD10,
Example 4.1] that by [COSW04, Example 11.5] the polynomial

$h_{F_{7}}+\mu x_{2}x_{4}x_{6}$

is stable only for $\mu=4$. As a side note we would like to mention that this
implies in particular that

$\sum_{S\in\binom{[7]}{3}}|\det(A[S])|\cdot\prod_{i\in S}x_{i}$

is not stable, giving a negative answer to the question raised in [Pur18,
Remark 4.2].

Now we are ready to disprove the following conjecture by Brändén–D’León.

######  Conjecture 4.9 ([BGD10, Conjecture 4.2]).

Suppose that $M$ has the weak half-plane property. Then so does any relaxation
of $M$.

Our counterexample is a suitable relaxation of $P_{8}$.

######  Example 4.10.

Consider again the matroid $P_{8}$ from 4.5. The set $X:=\\{3,5,6,7\\}$ is a
circuit-hyperplane of $P_{8}$. Following [MR08] we denote by $P_{1}$ the
relaxation of $P_{8}$ by $X$. We will prove that $P_{1}$ does not have the
weak half-plane property, although it is a relaxation of $P_{8}$. Using the
Macaulay2 [GS] package “Matroids” [Che18] we compute that
$\dim(V_{P_{1}})=10$. Thus by 4.5 and 4.7 the vectors $\delta_{X}$ and
$\iota(u(A))$ span a linear complement of $W_{P_{1}}$ in $V_{P_{1}}$. We
define $v=\frac{1}{\log(2)}\iota(u(A))$ — this is just for convenience to get
a vector with entries in $\\{0,1,2\\}$. If $P_{1}$ has the weak half-plane
property, then by 4.3 there are $a,b>0$ such that

$F_{a,b}=\sum_{B\text{ basis of }P_{8}}b^{v_{B}}x^{B}+ax_{3}x_{5}x_{6}x_{7}$

is stable. Now consider the $(0,1)$-th Rayleigh difference

$\Delta_{0,1}F_{a,b}:=\partial_{x_{0}}F_{a,b}\partial_{x_{1}}F_{a,b}-F_{a,b}\partial_{x_{0}}\partial_{x_{1}}F_{a,b}.$

By [Brä07, Theorem 5.10] the Rayleigh difference $\Delta_{0,1}F_{a,b}$ is
globally nonnegative if $F_{a,b}$ is stable. Plugging in $(1,1,t,-1,-1,t)$ for
$(x_{2},\ldots,x_{8})$ yields

$(\Delta_{0,1}F_{a,b})(1,1,t,-1,-1,t)=-abt^{3}+(-ab-4b^{2}+2a+12b+16)t^{2}+at.$

Because $a,b>0$, this takes negative values for large enough $t$. Therefore,
for no choice of $a,b>0$ the polynomial $F_{a,b}$ is stable.

######  Remark 4.11.

According to [MR08, Proposition 4] the matroid $P_{1}$ is not representable
over any field. It does not have the half-plane property because it has the
matroid $F_{7}^{-3}$ as a minor which does not have the half-plane property by
[COSW04, Example 11.7]. Moreover, it does not seem to be representable even in
the more general context studied in [PVZ13]. This made $P_{1}$ a good
candidate for being a counterexample to 4.9 as representations over certain
algebras (that are not necessarily fields) sometimes can still be used to
prove the weak half-plane property. See for instance [AB18] where the weak
half-plane property was proved for the non-Pappus and the non-Desargues
matroid.

######  Remark 4.12.

We use the notation as in 4.10. The polynomial $F_{0,1}:=\lim_{a\to 0}F_{a,1}$
is the basis generating polynomial of $P_{8}$. The polynomial
$F_{0,0}:=\lim_{a,b\to 0}F_{a,b}$ is the basis generating polynomial of the
graphical matroid $M(G_{1})$ where $G_{1}$ is depicted in Figure 2 on the
left. In total, the regular subdivision of the matroid polytope of $P_{8}$
defined by $v_{B}$ has six maximal cells, all of which are matroid polytopes
themselves. Four of the corresponding matroids are isomorphic to $M(G_{1})$,
the two remaining ones are isomorphic to $M(G_{2})$ where $G_{2}$ is the graph
on the right of Figure 2. Unfortunately, this knowledge did not help us
representing $F_{a,b}$ in a simple way.

61524703

Figure 2. The graphs $G_{1}$ (left) and $G_{2}$ (right) from 4.12.

##  5\. Quaternionic unimodular matroids

We recall the definition of quaternionic unimodular (QU) matroids. To this
end, let ${\mathbb{H}}$ denote the skew field of quaternions.

######  Definition 5.1.

Let $E$ be a finite set and $C\subseteq{\mathbb{H}}^{E}$ a submodule of the
free left ${\mathbb{H}}$-module ${\mathbb{H}}^{E}$. A nonzero element $x\in C$
is called an _elementary chain_ of $C$ if $C$ does not contain a nonzero
element whose support is strictly contained in the support of $x$. The
submodule $C$ is called _unimodular_ if for every elementary chain $x$ of $C$
all nonzero entries of $x$ have the same norm.

######  Theorem 5.2 ([PVZ13, Theorem 3.7]).

Let $E$ be a finite set and $C\subseteq{\mathbb{H}}^{E}$ unimodular. The set
of supports of elementary chains in $C$ is the set of cocircuits of a matroid
$M(C)$ on $E$.

######  Definition 5.3.

A matroid on a finite set $E$ of the form $M(C)$ for some unimodular
$C\subseteq{\mathbb{H}}^{E}$ is called _quaternionic unimodular (QU)_.

The goal of this section is to prove that every QU matroid has the half-plane
property. This has been conjectured in [PVZ13, Conjecture 6.9].

######  Definition 5.4 ([PVZ13, page 219]).

Denote by $\varphi\colon{\mathbb{H}}\to{\mathbb{C}}^{2\times 2}$ the map

$\varphi(a+bi+cj+dk)=\begin{pmatrix}a+bi&c+di\\\ -c+di&a-bi\end{pmatrix}.$

We extend $\varphi$ to matrices by applying $\varphi$ entry-wise. Thus for a
matrix $A\in{\mathbb{H}}^{n\times n}$ we obtain the $2n\times 2n$ complex
matrix $\varphi(A)$. We define

$\delta(A):=\sqrt{|\det(\varphi(A))|}.$

######  Remark 5.5.

We collect some basic properties of $\delta$.

  1. (1)

For columns $a_{1},\ldots,a_{n}\in{\mathbb{H}}^{n}$ and
$\lambda\in{\mathbb{R}}$, we have

$\delta(\lambda a_{1}\ a_{2}\ldots a_{n})=|\lambda|\cdot\delta(a_{1}\ldots
a_{n}).$

This follows directly from the definition.

  2. (2)

For matrices $A,B\in{\mathbb{H}}^{n\times n}$ we have
$\delta(AB)=\delta(A)\delta(B)$ and $\delta(A)=\delta(A^{*})$. This is [PVZ13,
Lemma 5.2].

For a matrix $A\in{\mathbb{H}}^{m\times n}$ with $m\leq n$ and $B\subseteq[n]$
of size $m$ we denote by $A[B]$ the $m\times m$ submatrix of $A$ consisting of
the columns indexed by $B$. The following is a version of the Cauchy–Binet
theorem over ${\mathbb{H}}$.

######  Proposition 5.6 ([PVZ13, Theorem 5.1]).

Let $A\in{\mathbb{H}}^{m\times n}$ be a matrix over the quaternions and $m\leq
n$. Then

$\delta(AA^{*})=\sum_{\begin{subarray}{c}B\subseteq[n]\\\
|B|=m\end{subarray}}\delta(A[B]A[B]^{*}).$

######  Lemma 5.7.

Let $E$ be a finite set and $C\subseteq{\mathbb{H}}^{E}$ unimodular. Let $d$
be the rank of the matroid $M=M(C)$. There is a $d\times|E|$ matrix $A$ over
${\mathbb{H}}$ whose rows form a basis of $C$ such that for all $B\subseteq E$
of size $d$ we have $\delta(A[B])=1$ if $B$ is a basis of $M$ and
$\delta(A[B])=0$ otherwise.

###### Proof.

Let $A$ be a matrix over ${\mathbb{H}}$ whose rows form a basis of $C$. By
[PVZ13, Lemma 3.14] it has $d$ rows, so $A\in{\mathbb{H}}^{d\times|E|}$. After
multiplying $A$ from the left by an invertible $d\times d$ matrix over
${\mathbb{H}}$, we can assume by [PVZ13, Corollary 3.26] that $A$ is a _strong
QU matrix_ in the sense of [PVZ13, Definition 3.23]. Now the statement of the
lemma follows from [PVZ13, Claim 5.4.1]. ∎

######  Theorem 5.8 ([PVZ13, Conjecture 6.9]).

Every QU matroid has the half-plane property.

###### Proof.

Let $E=\\{1,\ldots,m\\}$ and $C\subseteq{\mathbb{H}}^{m}$ be unimodular.
Denote by $M=M(C)$ the associated QU matroid and let
$A\in{\mathbb{H}}^{d\times m}$ be a matrix as in 5.7. We have to show that the
bases generating polynomial $h_{M}$ of $M$ is stable. This is equivalent to
$h_{M}^{2}$ being stable. We prove this by showing that $h_{M}^{2}$ agrees
with the stable polynomial

$\det\left(\varphi(A)\begin{pmatrix}x_{1}&&&&\\\ &x_{1}&&&\\\ &&\ddots&&\\\
&&&x_{m}&\\\ &&&&x_{m}\end{pmatrix}\varphi(A)^{*}\right).$

It suffices to show that these two polynomials agree on the positive orthant.
We denote by $a_{1},\ldots,a_{m}\in{\mathbb{H}}^{d}$ the columns of $A$. Let
$x\in{\mathbb{R}}_{>0}^{m}$ and write $x=(x_{1}^{2},\ldots,x_{m}^{2})$ for
$x_{1},\ldots,x_{m}\in{\mathbb{R}}_{>0}$. Then we have

(2) $\displaystyle h_{M}(x)^{2}$
$\displaystyle=\left(\sum_{\begin{subarray}{c}B\subseteq E\\\
|B|=d\end{subarray}}\delta(A[B]A[B]^{*})x^{B}\right)^{2}$ (3)
$\displaystyle=\left(\sum_{\begin{subarray}{c}B\subseteq E\\\
|B|=d\end{subarray}}\delta((x_{1}a_{1}\ldots x_{m}a_{m})[B](x_{1}a_{1}\ldots
x_{m}a_{m})[B]^{*})\right)^{2}$ (4) $\displaystyle=\delta((x_{1}a_{1}\ldots
x_{m}a_{m})(x_{1}a_{1}\ldots x_{m}a_{m})^{*})^{2}$ (5)
$\displaystyle=|\det(\varphi(x_{1}a_{1}\ldots
x_{m}a_{m})\varphi(x_{1}a_{1}\ldots x_{m}a_{m})^{*})|$ (6)
$\displaystyle=|\det\left(\varphi(A)\begin{pmatrix}x_{1}^{2}&&&&\\\
&x_{1}^{2}&&&\\\ &&\ddots&&\\\ &&&x_{m}^{2}&\\\
&&&&x_{m}^{2}\end{pmatrix}\varphi(A)^{*}\right)|$ (7)
$\displaystyle=\det\left(\varphi(A)\begin{pmatrix}x_{1}^{2}&&&&\\\
&x_{1}^{2}&&&\\\ &&\ddots&&\\\ &&&x_{m}^{2}&\\\
&&&&x_{m}^{2}\end{pmatrix}\varphi(A)^{*}\right).$

Here we have equality in (2) by 5.5 and 5.7. (3) holds by 5.5. For (4) we use
5.6 and (5) follows from the definition of $\delta$. (6) and (7) are obvious.
∎

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