Sign conditions for the existence of at least one positive solution of a sparse polynomial system

TL;DR

This paper establishes sign-based criteria on the support and coefficients of sparse polynomial systems that ensure at least one positive real solution, using degree theory and Gale duality, with connections to toric ideals.

Contribution

It introduces new sign conditions for positive solutions of sparse polynomial systems, linking algebraic and geometric methods like Gale duality and toric ideals.

Findings

Sign conditions guarantee positive solutions in sparse systems

Conditions relate to Gale duality and degree theory

Connections made to algebraic properties of toric ideals

Abstract

We give sign conditions on the support and coefficients of a sparse system of d generalized polynomials in d variables that guarantee the existence of at least one positive real root, based on degree theory and Gale duality. In the case of integer exponents, we relate our sufficient conditions to algebraic conditions that emerged in the study of toric ideals.

Equations

$$f_i(x)=\sum _{j=1}^n{c}_{ij}{x}^{a_j}=0,i=1,\dots ,d.$$

$$A=\left(\begin{array}{ccc}1& \dots & 1\\ a_1& \dots & a_n\end{array}\right).$$

$${\mathcal{C}}^{\circ }={\mathbb{R}}_{>0}{C}_1+\cdots +{\mathbb{R}}_{>0}{C}_n,$$

$$0\in {\mathcal{C}}^{\circ }.$$

$${\mathcal{C}}_P={\mathbb{R}}_{>0}{P}_1+\cdots +{\mathbb{R}}_{>0}{P}_n,$$

$${\mathcal{C}}_P^{\nu }=\{y\in {\mathbb{R}}^{k+1}:⟨P_i,y⟩>0,i=1,\dots ,n\},$$

$${\overline{\mathcal{C}}}_P={\mathbb{R}}_{\ge 0}{P}_1+\cdots +{\mathbb{R}}_{\ge 0}{P}_n.$$

$${\Delta }_P={\mathcal{C}}_P^{\nu }\cap \{y\in {\mathbb{R}}^{k+1}:y_0=1\}.$$

$$\prod _{i=1}^n⟨P_i,y{⟩}^{b_{ij}}=1,j=1,\dots ,k,$$

$$\begin{array}[]{rl}\varphi\colon\mathbb{R}^{k+1}&\to\mathbb{R}^{d}\\ z&\mapsto\left((D\cdot z)^{\alpha_{1}},\ldots,(D\cdot z)^{\alpha_{d}}\right),\end{array}$$

$$\mathrm{deg}(h,U,0)=(-1{)}^k.$$

$$g_j(y)=0,j=1,\dots ,k,g_j(y)=\prod _{b_{ij}>0}{p}_i(y{)}^{b_{ij}}-\prod _{b_{ij}<0}{p}_i(y{)}^{-b_{ij}},$$

$$p_i(y)=⟨P_i,(1,y)⟩.$$

$$g=(g_1,\dots ,g_k):{\mathbb{R}}^k\to {\mathbb{R}}^k.$$

$$\mathcal{F}({\Delta }_P)=\{L(F):F\text{mbox}isaproperfaceof{\Delta }_P\}$$

$$g_j(x)=-\prod _{b_{\ell j}<0}{p}_{\ell }(x{)}^{-b_{\ell j}},$$

$$g_j(x)=-\prod _{b_{\ell j}<0}{p}_{\ell }(x{)}^{-b_{\ell j}}.$$

$${\bar{b}}_{rj}:=\sum _{\ell \in K_r}{b}_{\ell j}.$$

$$\bar{L}=\{r\in \{1,\dots ,s\}:L\cap K_r\ne \varnothing \}$$

$$\prod _{r\in \{1,\dots ,s\}}⟨P_{i(r)},y{⟩}^{{\bar{b}}_{r,j}}=c_j,j=1,\dots ,k.$$

$${\bar{g}}_j(y)=0,j=1,\dots ,k,$$

$${\bar{g}}_j(y)=\prod _{{\bar{b}}_{r,j}>0}{p}_{i(r)}(y{)}^{{\bar{b}}_{r,j}}-c_j\prod _{{\bar{b}}_{r,j}<0}{p}_{i(r)}(y{)}^{-{\bar{b}}_{r,j}}.$$

$$C=\left(\begin{array}[]{ccccc}-1&-1&1&1&0\\ -(3c+8)&-c&2c+8&0&2\end{array}\right),$$

$$\begin{array}[]{rcl}-y^{4}-x^{5}y^{4}+x^{2}y^{8}+x^{3}&=&0,\\ -(3c+8)y^{4}-cx^{5}y^{4}+(2c+8)x^{2}y^{8}+2x^{3}y^{5}&=&0,\end{array}$$

$$B=\left(\begin{array}[]{rr}1&0\\ 2&1\\ 1&2\\ 0&1\\ -4&-4\end{array}\right)\qquad D=\left(\begin{array}[]{r|rr}1&1&0\\ 1&1&2\\ 1&2&1\\ 1&0&1\\ c&-4&-4\end{array}\right)$$

$$\left(\begin{array}{ccc}+& +& +\\ +& 0& -\\ +& -& 0\\ -& 0& 0\end{array}\right)$$

$$0\in {\mathbb{R}}_{>0}{\tilde{P}}_1+\cdots +{\mathbb{R}}_{>0}{\tilde{P}}_n,$$

$$B=\left(\begin{array}[]{cc}B_{v}&R\\ 0&I_{n-s}\end{array}\right),$$

$$\bar{C}=\left(\begin{array}{ccc}1& \cdots & 1\\ & C& \end{array}\right)$$

$$I_{\mathcal{L}}=⟨x^{u^{+}}-x^{u^{-}}:u\in \mathcal{L}⟩\subset \mathbb{R}[x_1,\dots ,x_n],$$

Full text

FB is partially supported by the grant ANR-18-CE40-0009 of Agence Nationale de Recherche, France. AD and MG are partially supported by UBACYT 20020170100048BA, CONICET PIP 11220150100473, and ANPCyT PICT 2016-0398, Argentina.

1. Introduction

Deciding whether a real polynomial system has a positive solution is a basic question, that is decidable via effective elimination of quantifiers [1]. There are few results on lower bounds on the number of real or positive roots of polynomial systems (see e.g. [3, 18, 19, 23]). In this paper, we consider generalized polynomial systems, that is, polynomials with real exponents, for which the positive solutions are well defined. We give sign conditions on the support and on the coefficients of a sparse system of $d$ generalized polynomials in $d$ variables that guarantee the existence of at least one positive real root, based on degree theory and Gale duality.

We fix an exponent set ${\mathcal{A}}=\{a_{1},\ldots,a_{n}\}\subset\mathbb{R}^{d}$ of cardinality $n$ and for any given real matrix $C=(c_{ij})\in\mathbb{R}^{d\times n}$ we consider the associated sparse generalized multivariate polynomial system in $d$ variables $x=(x_{1},\dots,x_{d})$ with support ${\mathcal{A}}$:

[TABLE]

We are interested by the existence of the positive solutions of (1.1) in $\mathbb{R}_{>0}^{d}$. Denoting by $n_{\mathcal{A}}(C)$ the (possibly infinite) number of positive real solutions of the system (1.1), our main goal is to give sufficient conditions on the exponent set ${\mathcal{A}}$ and the coefficient matrix $C$ that ensure $n_{\mathcal{A}}(C)>0$.

When ${\mathcal{A}}\subset\mathbb{Z}^{d}$ we consider the existence of solutions in the real torus $(\mathbb{R}^{*})^{d}$ of points in $\mathbb{R}^{d}$ with nonzero coordinates, and we relate our conditions to well-studied algebraic properties of lattice ideals associated with the configuration ${\mathcal{A}}$.

In applications, for example, in the context of chemical reaction networks, lower bounds of positive roots of polynomial systems guarantee the existence of (stoichiometrically compatible) positive steady states. In [13], sign conditions are used to decide if a family of polynomial systems associated with a given reaction network cannot admit more than one positive solution for any choice of the parameters and, in this case, conditions for the existence of one positive solution are given as a corollary of a result from [15], based on degree theory. Our point of view of searching for conditions on the exponent and the coefficient matrices of the system comes from this paper. As we do not assume injectivity (at most one root), we cannot use tools from these papers or the more recent article [14], as Hadamard’s theorem.

In [6] the authors use degree theory in the study of chemical reaction networks to describe parameters for which there is a single positive solution or for which there are more (this is called multistationarity). We apply some of these techniques in a Gale duality setting, more precisely, based on Theorem 3.1, which is version of a particular case of Theorem $2$ in the Supplementary Information of [6].

We can use different convex sets to apply Theorem 3.1. The first one which comes in mind is the positive orthant, which is not bounded. Another natural idea is to consider the Newton polytope of the polynomials in the system, or some dilates of it. This is reasonable since it is completely determined by the monomials appearing in the system. In this paper, we use another convex polytope which seems natural since it is determined by the coefficients of the system. We define this polytope (called $\Delta_{P}$) in § 2.3 using the Gale duality trick for polynomial systems that was studied by Bihan and Sottile in [4], see also [5]. We can think of $\Delta_{P}$ as a “shadow” of the positive orthant via Gale duality, which has the advantage that it can be chosen to be bounded.

In the case of a generalized polynomial $f$ in one variable, the existence of at least one positive real root based on degree theory is just the condition that the coefficients of the smallest exponent and the biggest exponent in $f$ have opposite signs. In case both coefficients have the same sign, this application is silent and other methods, in general ad hoc, are required to ensure a positive root. In our case, this is similar. We explicitly show in Example 3.9 that when we restrict our results to the case when ${\mathcal{A}}\subset\mathbb{R}^{n}$ is a circuit, our sufficient conditions to have $n_{\mathcal{A}}(C)>0$ are indeed equivalent to the fact that the two extremal coefficients in the Descartes type sign variation bound for $n_{\mathcal{A}}(C)$ in [2] have opposite signs. In the general case, we get interesting and more involved sufficient sign conditions relating the associated exponent matrix $A$ in (2.1) and the coefficient matrix $C$, but these conditions are not necessary and other tools are needed when they do not hold.

In Section 2 we recall the notion of Gale duality and the basic duality of solutions (see Theorem 2.5), and we introduce useful notation as well as the necessary condition (2.3) of the existence of a positive vector in ${\rm ker}(C)$ for $n_{A}(C)$ being positive, which only depends on the signs of the maximal minors of $C$ (that is, on its associated oriented matroid).

In Section 3, we recall the basic concepts of degree theory and we present our main result Theorem 3.8, which gives conditions on the Gale duality side to guarantee the existence of positive solutions. This result has some technical hypotheses that relate the lattice of faces of $\Delta_{P}$ plus signs of entries in a Gale dual matrix of the coefficient matrix $C$ of the system, with sign conditions on a Gale dual of the matrix $A$ of exponents. We present Example 3.10 to clarify our definitions and to show that the sufficient conditions in Theorem 3.8 could imply the existence of more than just one positive solution.

In the following sections we give sufficient conditions on the support and the matrix of coefficients that ensure that Theorem 3.8 can be applied. In Section 4, we consider the notion of mixed dominating matrices from [11] (see Definition 4.1) to get Theorem 4.6. Under the hypotheses of this result, we show that $n_{A}(C)>0$ whenever the vectors of signs of a basis of ${\rm ker}(A)$, can be realized as vectors of signs of elements in ${\rm ker}(C)$, which is also a condition only depending on the associated oriented matroid of $C$. In Section 5, we give geometric conditions on ${\mathcal{A}}$ and $C$ that guarantee that the hypotheses of Theorem 3.8 are satisfied. We introduce for this the notion of $I$-compatibility in Definition  5.6, which is clarified in Example 5.7 and Figure 2. Based on results from [10], we get Theorem 5.8 which guarantees $n_{A}(C)>0$ in terms of the relative combinatorial positions of the configuration of exponents and the configuration of the points given by the columns of $C$.

In the last Section 6, we concentrate our study on integer configurations ${\mathcal{A}}$. We relate the dominance conditions to algebraic conditions that emerged in the study of toric ideals, and we naturally extend in this case our approach to ensure the existence of solutions in the real torus $(\mathbb{R}^{*})^{d}$. Our last result is Theorem 6.10, which shows which sign conditions can be attained by the corresponding Gale dual system (2.8).

2. Gale duality for positive solutions of polynomial systems

We first present basic definitions and results on Gale duality. Given a matrix $M\in{\mathbb{R}^{r\times s}}$ of maximal rank $r$, a Gale dual matrix of $M$ is any matrix $N\in\mathbb{R}^{s\times(s-r)}$ of maximal rank whose columns vectors are a basis of the kernel of $M$. Clearly a Gale dual matrix is not unique as it corresponds to a choice of a basis: it is unique up to right multiplication by an invertible $(s-r)\times(s-r)$-matrix. We will also say that the $s$ row vectors of $N$ define a Gale dual configuration in $\mathbb{R}^{s-r}$ to the configuration in $\mathbb{R}^{r}$ defined by the $s$ column vectors of $M$. We will introduce a Gale dual system (2.8) and polyhedra $\Delta_{P}$ (2.7), depending on the choice of a Gale dual matrix to the coefficient matrix $C$. We will then recall Theorem 2.5, which gives a fundamental link between the positive real roots of system (1.1) and the solutions in $\Delta_{P}$ of the Gale dual system (2.8).

2.1. Matrices and their Gale duals

Let ${\mathcal{A}}=\{a_{1},\ldots,a_{n}\}$ be a finite subset of $\mathbb{R}^{d}$ of cardinality $n$ and $C=(c_{ij})\in\mathbb{R}^{d\times n}$. As we mentioned in the introduction, we are interested in the solvability of the associated sparse generalized multivariate polynomial system (1.1) in $d$ variables $x=(x_{1},\dots,x_{d})$ with support ${\mathcal{A}}$ and coefficient matrix $C$.

Note that if we multiply each equation of system (1.1) by a monomial (i.e, we translate the configuration $\mathcal{A}$), the number of positive real solutions does not change, and then $n_{\mathcal{A}}(C)$ is an affine invariant of the point configuration $\mathcal{A}$. It is then natural to consider the matrix $A\in\mathbb{R}^{(d+1)\times n}$ with columns $(1,a_{1}),(1,a_{2}),\ldots,(1,a_{n})\in\mathbb{R}^{d+1}:$

[TABLE]

We will refer to the matrix $A$ as the corresponding matrix of the point configuration $\mathcal{A}$.

We will always assume that $C$ is of maximal rank $d$ and $A$ is of maximal rank $d+1$. Then, we need to have $n\geq d+1$. If equality holds, it is easy to see that system (1.1) has a positive solution if and only if the necessary condition (2.3) holds. So we will suppose that $n\geq d+2$.

We denote by $k=n-d-1$ the codimension of $A$ (and of ${\mathcal{A}}$). Note that the codimension of $C$ equals $k+1$. Let $B=(b_{ij})\in\mathbb{R}^{n\times k}$ be a matrix which is Gale dual to $A$, and let $D=(d_{ij})\in\mathbb{R}^{n\times(k+1)}$ be any a matrix which is Gale dual to $C$. We will number the columns of $B$ from 1 to $k$ and the columns of $D$ from [math] to $k$ and denote by $P_{1},\dots,P_{n}\in\mathbb{R}^{k+1}$ the row vectors of $D$, that is, the Gale dual configuration to the columns of $C$.

2.2. A necessary condition

There is a basic necessary condition for $n_{\mathcal{A}}(C)$ to be positive. Denote by $C_{1},\dots,C_{n}\in\mathbb{R}^{d}$ the column vectors of the coefficient matrix $C$ and call

[TABLE]

the positive cone generated by them. Given a solution $x\in\mathbb{R}_{>0}^{d}$ of system (1.1), the vector $(x^{a_{1}},\dots,x^{a_{n}})$ is positive and so the origin ${\bf 0}\in\mathbb{R}^{d}$ belongs to $\mathcal{C}^{\circ}$. Then, necessarily

[TABLE]

It is a well-known result that Condition (2.3) holds if and only if the vectors $P_{1},\dots,P_{n}$ lie in an open halfspace through the origin. Indeed, any vector in the kernel of $C$ is of the form $(\langle P_{1},u\rangle,\dots,\langle P_{n},u\rangle)$, and so there exists a positive vector in the kernel if and only there exists a vector $u\in\mathbb{R}^{k+1}$ such that $\langle P_{i},u\rangle>0$ for any $i$. Note that Condition (2.3), together with the hypothesis that $C$ is of maximal rank $d$, is equivalent to ${\mathcal{C}}^{\circ}=\mathbb{R}^{d}$.

2.3. Defining cones and polytopes in Gale dual space

We also define other cones that we will use. Denote by

[TABLE]

the positive cone generated by the rows of a Gale dual matrix $D$ and let

[TABLE]

be its dual open cone. Note that if $C$ has maximal rank $d$ and Condition (2.3) holds, the cone ${\mathcal{C}}_{P}$ is strictly convex. Therefore, its dual open cone ${\mathcal{C}}_{P}^{\nu}$ is a nonempty full dimensional open convex cone. We will also consider the closed cone

[TABLE]

The following Lemma is straightforward.

Lemma 2.1**.**

Assume that $C$ has maximal rank $d$ and that ${\bf 0}\in\mathcal{C}^{\circ}$. Then for any nonzero $u\in\overline{{\mathcal{C}}}_{P}$ and any $c\in\mathbb{R}_{>0}$, the polytope ${\mathcal{C}}_{P}^{\nu}\cap\{y\in\mathbb{R}^{k+1}\,:\,\langle u,y\rangle=c\}$ has dimension $k$. Moreover, this polytope is bounded if and only $u\in{\mathcal{C}}_{P}$.

Define

[TABLE]

Corollary 2.2**.**

Assume that $C$ has maximal rank, ${\bf 0}\in\mathcal{C}^{\circ}$ and let $D$ be a Gale dual matrix of $C$. Then $(1,0,\ldots,0)\in{\mathcal{C}}_{P}$ if and only if $\Delta_{P}$ has dimension $k$ and is bounded.

We next show that we can always find a Gale matrix $D$ such that $\Delta_{P}$ is nonempty and bounded.

Lemma 2.3**.**

Assume that $C$ has maximal rank. Then there is a Gale dual matrix $D$ of $C$ such that $(1,0,\ldots,0)\in{\mathcal{C}}_{P}$.

Proof.

Start with any Gale dual matrix $D$ of $C$ and pick any vector $u\in{\mathcal{C}}_{P}$. Then there is an invertible matrix $R\in\mathbb{R}^{(k+1)\times(k+1)}$ such that $u\cdot R=(1,0,\ldots,0)$, where $u$ is written as a row vector. Consider the matrix $D^{\prime}=DR$ and denote by $P_{1}^{\prime},\ldots,P_{n}^{\prime}$ its row vectors. Then $D^{\prime}$ is Gale dual to $C$, and $(1,0,..,0)\in{\mathcal{C}}_{P^{\prime}}=\mathbb{R}_{>0}P^{\prime}_{1}+\cdots+\mathbb{R}_{>0}P^{\prime}_{n}$. ∎

To any choice of Gale dual matrices $B$ and $D$ of $A$ and $C$ respectively, we associate the following system with unknowns $y=(y_{0},\ldots,y_{k})$:

[TABLE]

which is called a Gale dual system of (1.1). Denote $G_{j}(y)=\prod_{i=1}^{n}\langle P_{i},y\rangle^{b_{ij}}$. Another choice $D^{\prime}$ of a Gale dual matrix for $C$ corresponds to another choice $y^{\prime}$ of linear coordinates for $\mathbb{R}^{k+1}$: if $D^{\prime}=DR$ with $R\in\mathbb{R}^{(k+1)\times(k+1)}$ invertible, then setting $y^{\prime}=R^{-1}(y)$ we get $D^{\prime}y^{\prime}=Dy$, where $y$ and $y^{\prime}$ as considered as column vectors. Another choice $B^{\prime}$ of a Gale dual matrix for $A$ gives an equivalent Gale system $H_{1}=\dots=H_{k}=1$, where for each $j$ there exist exponents $(\mu_{1},\dots,\mu_{k})$ such that $H_{j}=G_{1}^{\mu_{1}}\dots G_{k}^{\mu_{k}}$.

Note that (2.8) is homogeneous of degree zero since the columns of $B$ sum up to zero. For any cone ${\mathcal{C}}\in\mathbb{R}^{n}$ with apex the origin, its projectivization $\mathbb{P}{\mathcal{C}}$ is the quotient space ${\mathcal{C}}/\sim$ under the equivalence relation $\sim$ defined by: for all $y,y^{\prime}\in{\mathcal{C}}$, we have $y\sim y^{\prime}$ if and only if there exists $\alpha>0$ such that $y=\alpha y^{\prime}$.

We will often use the following observation.

Remark 2.4**.**

If $(1,0,\ldots,0)\in\overline{{\mathcal{C}}}_{P}$, then ${\mathcal{C}}_{P}^{\nu}$ is contained in the open half-space defined by $y_{0}>0$. Thus, the map $(y_{0},y_{1},\ldots,y_{k})\mapsto(1,y_{1}/y_{0},\ldots,y_{k}/y_{0})$ induces a bijection between $\mathbb{P}{\mathcal{C}}_{P}^{\nu}$ and $\Delta_{P}$.

2.4. The equivalence of solutions

Here is a slight variation of Theorem 2.2 in [4].

Theorem 2.5**.**

There is a bijection between the positive solutions of the initial system (1.1) and the solutions of the Gale dual system (2.8) in $\mathbb{P}{\mathcal{C}}_{P}^{\nu}$, which induces a bijection between the positive solutions of  (1.1) and the solutions of (2.8) in $\Delta_{P}$ when $(1,0,\ldots,0)\in\overline{{\mathcal{C}}}_{P}$.

Proof.

If $x\in\mathbb{R}_{>0}^{d}$ is a solution of the system (1.1), then $(x^{a_{1}},\ldots,x^{a_{n}})$ belongs to $\ker(C)\cap\mathbb{R}_{>0}^{n}$. Thus, there exists $y\in\mathbb{R}^{k+1}$ (which is unique since $D$ has maximal rank) such that $x^{a_{i}}=\langle P_{i},y\rangle$ for $i=1,\ldots,n$. Then, $y\in{\mathcal{C}}_{P}^{\nu}$ and $y$ is a solution of the Gale dual system (2.8). If furthermore $(1,0,\ldots,0)\in\overline{{\mathcal{C}}}_{P}$, then dividing by $y_{0}$ if necessary, a solution $y\in{\mathcal{C}}_{P}^{\nu}$ of (2.8) gives a solution of the same system in $\Delta_{P}$ because it is homogeneous of degree zero. We showed in Remark 2.4 that the previous map is bijective by giving explicitly its inverse map.

Now, let $y\in{\mathcal{C}}_{P}^{\nu}$ be a solution of (2.8). Let $(e_{1},\ldots,e_{d})$ be the canonical basis of $\mathbb{R}^{d}$. Since $A$ has maximal rank, there exists $\alpha_{j}=(\alpha_{1j},\ldots,\alpha_{nj})\in\mathbb{R}^{n}$, for $j=1,\ldots,d$, such that $e_{j}=\sum_{i=1}^{n}\alpha_{ij}a_{i}$. To any column vector $z\in\mathbb{R}^{k+1}$, we associate the vector $D\cdot z$ with coordinates $\langle P_{i},z\rangle$, $i=1,\ldots,n$. Consider now the map

[TABLE]

where $(D\cdot z)^{\alpha_{j}}=\prod_{i=1}^{n}\langle P_{i},z\rangle^{\alpha_{ij}}$. Let $x=\varphi(y)$. Then, $x^{a_{i}}=\langle P_{i},y\rangle$ for $i=1,\ldots,n$, which gives $(x^{a_{1}},\ldots,x^{a_{n}})\in\ker(C)$. Moreover, since $y\in{\mathcal{C}}_{P}^{\nu}$, we have that $x\in\mathbb{R}_{>0}^{d}$, and then $x$ is a positive solution of system (1.1). ∎

Remark 2.6**.**

Theorem 2.2 in [4] is a particular case of Theorem 2.5 taking a Gale dual matrix $D$ with the identity matrix $I_{k+1}$ as a submatrix (in which case the condition that $(1,0,\ldots,0)\in\overline{{\mathcal{C}}}_{P}$ is trivially satisfied).

3. Existence of positive solutions via Gale duality and degree theory

In this section, we present Theorem 3.8, which gives conditions on the Gale dual matrices $B$ and $D$ that guarantee the existence of at least one positive solution of the system (1.1). As we mentioned in the Introduction, our results are based on degree theory. We will consider coefficient matrices $C$ of maximal rank $d$ such that the necessary condition (2.3) is satisfied. As before, we will fix a matrix $D=(d_{ij})\in\mathbb{R}^{n\times(k+1)}$ which is Gale dual to $C$ and denote by $P_{1},\dots,P_{n}\in\mathbb{R}^{k+1}$ the row vectors of $D$, which are all nonzero. We will moreover assume that the polytope $\Delta_{P}$ in  (2.7) is nonempty and bounded. Our main result is Theorem 3.8. We also point out how the statement simplifies in case $C$ is uniform, that is, when all maximal minors of $C$ are nonzero and we give examples that show how this result can be applied.

Given an open set $U\subset\mathbb{R}^{k}$, a function $h\in\mathcal{C}^{0}(\overline{U},\mathbb{R}^{k})$ and $y\in\mathbb{R}^{k}\setminus h({\partial}U)$, the symbol $\operatorname{deg}(h,U,y)$ denotes the Brouwer degree (which belongs to $\mathbb{Z}$) of $h$ with respect to $(U,y)$. A main result in degree theory is that if $\operatorname{deg}(h,U,y)\neq 0$, then there exists at least one $x\in U$ such that $y=h(x)$. For background and the main properties about Brouwer degree, we refer to Section 2 in the Supplementary Information of [6] and Section 14.2 in [22].

We present the version of the Brouwer’s theorem that we will use. This version is a particular case of Theorem $2$ in the Supplementary Information of [6] (here we take $W$ empty), and also appears in the proof of Lemma 2 of [7]. Recall that a vector $v\in\mathbb{R}^{k}$ points inwards $U\subset\mathbb{R}^{k}$ at a boundary point $x\in\partial U$, if for small $\varepsilon>0$ it holds that $x+\varepsilon v\in\overline{U}$.

Theorem 3.1** ([6, 7]).**

Let $h:\mathbb{R}^{k}\to\mathbb{R}^{k}$ be a $\mathcal{C}^{1}$-function. Let $U$ be an open, nonempty, bounded and convex subset of $\mathbb{R}^{k}$ such that

1. i)

$h(x)\neq 0$* for any $x\in\partial U$.* 2. ii)

for every $x\in\partial U$, the vector $h(x)$ points inwards $U$ at $x$.

Then,

[TABLE]

In particular, there exists a point $x$ in $U$ such that $h(x)=0$. Moreover, assuming the zeros are nondegenerate, if there exists a zero $x^{*}\in U$ where the Jacobian at $x^{*}$ has the same sign as $(-1)^{k+1}$, then there are at least three zeros and always an odd number.

Define the sign of any real number $r$ by $\operatorname{sign}(r)=+1,-1,0$ according as $r>0$, $r<0$ or $r=0$ respectively. The sign of any vector $r=(r_{1},\ldots,r_{k})\in\mathbb{R}^{k}$ is then defined by $\operatorname{sign}(r)=(\operatorname{sign}(r_{1}),\ldots,\operatorname{sign}(r_{k}))$.

In view of Theorem 2.5, we look for the solutions of (2.8) in $\Delta_{P}$. Plugging $y_{0}=1$ in (2.8) and clearing the denominators, we get a generalized polynomial system in $\Delta_{P}$ on variables $y=(y_{1},\ldots,y_{k})$:

[TABLE]

where

[TABLE]

We denote by $g$ the Gale map:

[TABLE]

Recall that Condition (2.3) implies that all $P_{i}$ lie in an open halfspace through the origin.

Definition 3.2**.**

Given $C\in\mathbb{R}^{d\times n}$ of maximal rank $d$ satisfying condition (2.3), we denote by $\bar{I}_{C}\subset\{1,\dots,n\}$ the set of indices of the vectors $P_{i}$ which belong to the boundary of the cone $\mathcal{C}_{P}$ and by $I_{C}\subset\bar{I}_{C}$ the set of indices of the vectors $P_{i}$ which lie on the rays of the cone (that is, on its one-dimensional faces).

As the facets of ${\mathcal{C}}_{P}^{\nu}$ are supported on the orthogonal hyperplanes $P_{i}^{\bot}$ for $i\in I_{C}$, for any $i\in I_{C}$ the vector $P_{i}$ is an inward normal vector of ${\mathcal{C}}_{P}^{\nu}$ at any point in the relative interior of the facet supported on $P_{i}^{\bot}$. It follows that the facets of the polytope $\Delta_{P}$ are supported on the hyperplanes $p_{i}(y)=0$ for $i\in I_{C}$, and that $(d_{i1},\ldots,d_{ik})$ is an inward normal vector of $\Delta_{P}$ at any point in the relative interior of the facet $F_{i}$. Also, we denote by $F_{i}$ the face of $\Delta_{P}$ supported on $p_{i}(y)=0$ for any $i\in\bar{I}_{C}$, and for any $L\subset\bar{I}_{C}$ we set $F_{L}=\cap_{i\in L}F_{i}$. Here, by a face of $\Delta_{P}$ we mean a face of the closure of $\Delta_{P}$. We denote by $F_{L}^{\circ}$ the relative interior of $F_{L}$.

Definition 3.3**.**

Let $C\in\mathbb{R}^{d\times n}$ of maximal rank $d$ satisfying condition (2.3), with Gale dual $D$. For a given proper face $F$ of $\Delta_{P}$, we denote by $L(F)=\{i\in\bar{I}_{C}\,:\,F\subset F_{i}\}$ the maximal set of indices $L\subset\bar{I}_{C}$ such that $F=F_{L}$ and set

[TABLE]

Note that $\bar{I}_{C}$ and $I_{C}$ depend on $C$ and are independent of the choice of Gale dual matrix $D$. Moreover, in case $C$ is uniform $D$ is also uniform, $\bar{I}_{C}=I_{C}$ and the cardinality of $L(F)$ equals the codimension of the face, which means that the polytope $\Delta_{P}$ is simple.

The sign of $g$ along the boundary of $\Delta_{P}$ can sometimes be determined as follows.

Lemma 3.4**.**

Let $A\in\mathbb{R}^{(d+1)\times n}$ as in (2.1) of maximal rank $d+1$, $C\in\mathbb{R}^{d\times n}$ a uniform matrix satisfying the necessary condition (2.3), and let $B\in\mathbb{R}^{n\times k}$ and $D\in\mathbb{R}^{n\times(k+1)}$ be Gale dual matrices of $A$ and $C$ respectively. Let $g=(g_{1},\dots,g_{k})$ the Gale map as in (3.3). Let $j\in\{1,\ldots,k\}$.

1. (1)

Let $F_{i}$ be any facet of $\Delta_{P}$ and let $x\in F_{i}^{\circ}$. If $b_{ij}\neq 0$, then $\operatorname{sign}(g_{j}(x))=-\operatorname{sign}(b_{ij})$. 2. (2)

Let $L\in\mathcal{F}(\Delta_{P})$ and $x\in F_{L}^{\circ}$. Assume that $\{b_{\ell j}\,:\,\ell\in L\}\neq\{0\}$.

1. (i)

If there exists $\ell_{0},\ell_{1}\in L$ such that $b_{\ell_{0}j}\cdot b_{\ell_{1}j}<0$, then $g_{j}(x)=0$. 2. (ii)

*If $b_{\ell j}\geq 0$ for all $\ell\in L$ then $\operatorname{sign}(g_{j}(x))=-1$, and if $b_{\ell j}\leq 0$ for all $\ell\in L$ then $\operatorname{sign}(g_{j}(x))=+1$. *

Proof.

Since $B$ is a Gale dual matrix of $A$, then each column of $B$ contains at least a positive and a negative entry, that is, the sets $\{b_{\ell j}>0\,:\ell=1,\dots,n\}$ and $\{b_{\ell j}<0\,:\ell=1,\dots,n\}$ are nonempty for each $j=1,\dots,k$.

For the first item, note that the facet $F_{i}$ is supported on $p_{i}(y)=0$. Suppose $x\in F_{i}^{\circ}$ and $b_{ij}\neq 0$. We compute $g_{j}$ as in (3.1). If $b_{ij}>0$, then

[TABLE]

because $p_{i}(x)=0$. In this case, $g_{j}(x)<0$ because as $C$ is uniform, $D$ is also uniform and so no other $p_{\ell}$ can be a nonzero multiple of $p_{i}$. Then, $\operatorname{sign}(g_{j}(x))=-\operatorname{sign}(b_{ij})$. The case $b_{ij}<0$ is analogous.

For the second part, assume that $L\in\mathcal{F}(\Delta_{P})$ and $x\in F_{L}^{\circ}$, with $\{b_{\ell j}\,:\,\ell\in L\}\neq\{0\}$. If there exists $\ell_{0},\ell_{1}\in L$ such that $b_{\ell_{0}j}\cdot b_{\ell_{1}j}<0$, we have that $p_{\ell_{1}}(x)=p_{\ell_{2}}(x)=0$, and then both terms $\prod_{b_{\ell j}>0}p_{\ell}(x)^{b_{\ell j}}$ and $\prod_{b_{\ell j}<0}p_{\ell}(x)^{-b_{\ell j}}$ are zero. Then $g_{j}(x)=0$. In the other case, if $b_{\ell j}\geq 0$ for all $\ell\in L$, then $b_{\ell j}>0$ for some $\ell\in L$ because we assume that not all vanish, and thus

[TABLE]

Note that if $x\in F_{L}^{\circ}$, $p_{\ell}(x)=0$ if and only if $\ell\in L$. Then, since $b_{\ell j}\geq 0$ for all $\ell\in L$, we have that $p_{\ell}(x)\neq 0$ for all $\ell$ such that $b_{\ell j}<0$, and so the product $-\prod_{b_{\ell j}<0}p_{\ell}(x)^{-b_{\ell j}}$ is negative. The case where $b_{\ell j}\leq 0$ for all $\ell\in L$ is analogous. ∎

Corollary 3.5**.**

Let $A\in\mathbb{R}^{(d+1)\times n}$ as in (2.1) of maximal rank $d+1$, $C\in\mathbb{R}^{d\times n}$ a uniform matrix of rank $d$ satisfying condition (2.3), and $B\in\mathbb{R}^{n\times k}$ and $D\in\mathbb{R}^{n\times(k+1)}$ Gale dual matrices of $A$ and $C$ respectively. Let $g$ be the Gale map (3.3) associated to $B$ and $D$. If $g(x)=0$ and $x\in F_{L}^{\circ}$ (so $L\in\mathcal{F}(\Delta_{P})$), then for $j=1,\ldots,k$, either $\{b_{\ell j}\,:\,\ell\in L\}=\{0\}$, or $\{b_{\ell j}\,:\,\ell\in L\}$ contains a (strictly) positive and a (strictly) negative element. In particular, if $g$ vanishes in the relative interior of a facet $F_{\ell}$ then the $\ell$-th row of $B$ contains only zero entries.

In order to state Theorem 3.8 we need to introduce the following definitions.

Definition 3.6**.**

Let $C\in\mathbb{R}^{d\times n}$ of maximal rank $d$. We define an equivalence relation on $\{1,\dots,n\}$ where two indices $i_{1},i_{2}$ are equivalent whenever $P_{i_{1}}$ and $P_{i_{2}}$ are proportional. We can then partition $\bar{I}_{C}$ into equivalence classes $K_{1},\dots,K_{s}$.

Given $B\in\mathbb{R}^{n\times k}$ we define an associated matrix $\bar{B}\in\mathbb{R}^{s\times k}$ with the following coefficients. For any $r=1,\dots,s$ and any $j=1,\ldots,k$ set

[TABLE]

For any $r\in\{1,\dots,s\}$ choose an index $i(r)$ such that $i(r)\in K_{r}$. Given any $L\subset\{1,\dots,n\}$ we set

[TABLE]

and denote by $\bar{B}_{\bar{L}}$ the submatrix of $\bar{B}$ given by the rows with indices in $\bar{L}$.

Note that when $C$ satisfies (2.3), if $P_{\ell}$ and $P_{i}$ are equivalent, there exists a positive constant $c$ such that $P_{i}=cP_{\ell}$. If moreover $C$ is uniform, each equivalence class $K_{i}$ consists of a single index (that we can assume to be $i$) and $\bar{b}_{ij}=b_{ij}$ for any $i$ in $\bar{I}_{C}=I_{C}$ and any $j\in\{0,\dots,k\}$.

Definition 3.7**.**

We say that a matrix $M$ is weakly mixed if any column of $M$ either has only zero entries, or contains a positive and a negative element.

We now present the main result of this section.

Theorem 3.8**.**

Let $\mathcal{A}=\{a_{1},\dots,a_{n}\}\subset\mathbb{R}^{d}$ and $C\in\mathbb{R}^{d\times n}$ of maximal rank $d$. Let $A\in\mathbb{R}^{(d+1)\times n}$ as in (2.1) of maximal rank $d+1$, and $B\in\mathbb{R}^{n\times k}$ and $D\in\mathbb{R}^{n\times(k+1)}$ Gale dual matrices of $A$ and $C$ respectively. Assume that ${\bf 0}\in\mathcal{C}^{\circ}$ and that $\Delta_{P}$ is a full dimensional bounded polytope.

Assume furthermore that the following conditions hold:

1. (1)

*For any $L\in\mathcal{F}(\Delta_{P})$ the submatrix $\bar{B}_{\bar{L}}$ is not weakly mixed, that is, $\bar{B}_{\bar{L}}$ has a nonzero column whose entries are all either nonpositive, or nonnegative. * 2. (2)

For any $i\in\bar{I}_{C}$ as in Definition 3.2, and $r$ such that $i\in K_{r}$, the following holds:

• •

$\bar{b}_{rj}\cdot d_{ij}\geq 0$* for any $j=1,\ldots,k$,*

• •

there exists $j\in\{1,\ldots,k\}$ such that $\bar{b}_{rj}\cdot d_{ij}>0$,

• •

for all $j\in\{1,\ldots,k\}$, if $\bar{b}_{rj}=0$ then $d_{ij}=0$.

Then $n_{\mathcal{A}}(C)>0$.

Proof.

In view of Theorem 2.5, we look for the solutions of (2.8) in $\Delta_{P}$, given by $G_{j}(y)=1$ for $j=1,\ldots,k$. For each $r\in\{1,\ldots,s\}$, choose one representative $i(r)$ in the equivalence class $K_{r}$. Under the assumption ${\bf 0}\in\mathcal{C}^{\circ}$, if $P_{i}$ and $P_{j}$ are proportional then $P_{j}=c\cdot P_{i}$ with $c>0$, as noticed before. Therefore, there exist constants $c_{j}>0$ for $j=1,\ldots,k$ such that (2.8) is equivalent to

[TABLE]

Plugging $y_{0}=1$ and clearing the denominators, system (2.8) has the same solutions as the following generalized polynomial system in $\Delta_{P}$ on variables $y=(y_{1},\ldots,y_{k})$:

[TABLE]

where

[TABLE]

Note that by construction, if $r,r^{\prime}\in\{1,\ldots,s\}$ are distinct, then $P_{i(r)}$ and $P_{i(r^{\prime})}$ are not proportional. Thus system (3.6) looks like a system (3.1) with the exception given by the positive constants $c_{j}$. It is then not difficult to see that, the conclusions of Lemma 3.4 and Corollary 3.5 hold for the map $\bar{g}=(\bar{g}_{1},\ldots,\bar{g}_{k})$ corresponding to the system (3.6). To simplify the notation, we assume from now on that $C$ is uniform, and so, as we remarked before, $I_{C}=\bar{I}_{C}$ and $\bar{b}_{ij}=b_{ij}$.

Since $\Delta_{P}$ is full dimensional and bounded, $(1,0,\dots,0)\in\mathcal{C}_{P}$. To prove that $n_{\mathcal{A}}(C)>0$, by Theorem 2.5 it is sufficient to show that the Gale system (3.6) has at least one solution in $\Delta_{P}$. First note that a vector $v\in\mathbb{R}^{k}$ points inwards $\Delta_{P}$ at a point $y$ contained in the relative interior of the facet $F_{i}$ supported on $P_{i}^{\bot}$ with $i\in I_{C}$ if and only if $\langle(d_{i1},\ldots,d_{ik}),v\rangle\geq 0$. More generally $v\in\mathbb{R}^{k}$ points inwards $\Delta_{P}$ at a point $y$ in the relative interior of a face $F_{L}$ ($L\in\mathcal{F}(\Delta_{P})$) if and only if $\langle(d_{\ell 1},\ldots,d_{\ell k}),v\rangle\geq 0$ for any $\ell\in L$, by a classical result of convex geometry. Assumption (1) ensures that $g$ does not vanish at $\partial\Delta_{P}$, by Corollary 3.5. By Lemma 3.4, condition (2) ensures that $-g$ points inwards $\Delta_{P}$ at each point $x$ in the relative interior of any facet $F_{i}$. Then $-g$ also points inwards $\Delta_{P}$ at any point $x$ in the relative interior of a face $F_{L}$. The result follows now from Theorem 3.1, taking $U=\Delta_{P}$ and $h=-g$. ∎

Example 3.9**.**

Consider the codimension one case $k=1$ (which is treated carefully in [2]). Then $B\in\mathbb{R}^{(d+2)\times 1}$ is a column matrix and its entries are the coefficients $\lambda_{1},\ldots,\lambda_{d+2}$ of a nontrivial affine relation on $\mathcal{A}$. Assume that $A$ is uniform (equivalently, assume that $\mathcal{A}$ is a circuit 111A point configuration $\mathcal{A}$ of $d+2$ points is a circuit if any subset of $d+1$ points of $\mathcal{A}$ is affinely independent.). Then, $B$ has no zero entry. Assume moreover that $C$ is uniform and that ${\bf 0}\in\mathcal{C}^{\circ}$. Then, there exists a Gale dual matrix $D$ such that $\Delta_{P}$ is a bounded interval of $\mathbb{R}$. Moreover, there exists a vector $\delta\in\mathbb{R}^{2}$ such that $\langle P_{i},\delta\rangle>0$ for $i=1,\ldots,d+2$, where $P_{1},\ldots,P_{d+2}\in\mathbb{R}^{2}$ are the row vectors of $D$. Let $\alpha:\{1,\ldots,d+2\}\rightarrow\{1,\ldots,d+2\}$ be the bijection such that all determinants $\mbox{det}\,(P_{\alpha_{i}},P_{\alpha_{i+1}})$ are positive for $i=1,\ldots,d+1$. Then, by Theorem 2.9 in [2], we have $n_{\mathcal{A}}(C)\leq\operatorname{signvar}(\lambda_{\alpha_{1}},\lambda_{\alpha_{2}},\ldots,\lambda_{\alpha_{d+2}})$ and moreover the difference is an even integer number (see Proposition 2.12 in [2]). The endpoints of the interval $\Delta_{P}$ are the roots of the two extremal polynomials $p_{\alpha_{1}}$ and $p_{\alpha_{d+2}}$, equivalently, $I_{C}=\{\alpha_{1},\alpha_{d+2}\}$. Now the Gale polynomial $g=g_{1}:\mathbb{R}\rightarrow\mathbb{R}$ (or its opposite) points inwards $\Delta_{P}$ at its vertices if and only if $\lambda_{\alpha_{1}}\cdot\lambda_{\alpha_{d+2}}<0$, which is equivalent to $\operatorname{signvar}(\lambda_{\alpha_{1}},\lambda_{\alpha_{d+2}})=1$. But, $\operatorname{signvar}(\lambda_{\alpha_{1}},\lambda_{\alpha_{2}},\ldots,\lambda_{\alpha_{d+2}})$ and $\operatorname{signvar}(\lambda_{\alpha_{1}},\lambda_{\alpha_{d+2}})$ have the same parity. Thus $g:\mathbb{R}\rightarrow\mathbb{R}$ (or its opposite) points inwards $\Delta_{P}$ at its vertices if and only if $n_{\mathcal{A}}(C)$ is odd by Proposition 2.12 in [2]. Therefore, in the circuit case the sufficient condition to have $n_{\mathcal{A}}(C)>0$ which is given by Theorem 3.8 is equivalent to $n_{\mathcal{A}}(C)$ being odd. Now, for any integer $d\geq 2$, it is not difficult to get examples of circuits ${\mathcal{A}}\subset\mathbb{R}^{d}$ and matrices $C$ such that $n_{\mathcal{A}}(C)$ is odd and is different from $1$. This shows that our sufficient condition does not imply $n_{\mathcal{A}}(C)=1$ in general, and thus is not equivalent to the condition given in [13] ensuring that $n_{\mathcal{A}}(C)=1$.

We now present an example with $k=d=2$ to illustrate Theorem 3.8.

Example 3.10**.**

Let $\mathcal{A}\subset\mathbb{Z}^{2}$ be the set of points $a_{1}=(0,4)$, $a_{2}=(5,4)$, $a_{3}=(2,8)$, $a_{4}=(3,0)$ and $a_{5}=(3,5)$. Consider the matrix of coefficients

[TABLE]

where $c\in\mathbb{R}$ is a parameter. The polynomial system of two polynomial equations and two variables $x,y$:

[TABLE]

has support $\mathcal{A}$ and coefficient matrix $C$. Let $A$ as in (2.1). Choose the following Gale dual matrices of $A$ and $C$ :

[TABLE]

Then $p_{1}(y)=1+y_{1}$, $p_{2}(y)=1+y_{1}+2y_{2}$, $p_{3}(y)=1+2y_{1}+y_{2}$, $p_{4}(y)=1+y_{2}$ and $p_{5}(y)=c-4y_{1}-4y_{2}$. If $c>0$, the convex polytope $\Delta_{P}$ is nonempty, bounded and it has five facets supported on the lines $p_{i}=0$ for $i=1,\ldots,5$, see Figure 1. Moreover, if $c>0$, then the assumptions of Theorem 3.8 are satisfied and thus $n_{\mathcal{A}}(C)>0$.

We use Singular [8], a free software of computer algebra system, to check what happens when we vary the value of $c>0$.

%LIB "signcond.lib"; ring r=0, (c,x,y,t), dp; poly f1=-y^4-x^5y^4+x^2y^8+x^3; poly f2=-(3c+8)y^4-cx^5y^4+(2c+8)x^2y^8+2x^3y^5; ideal i=f1,f2, diff(f1,x)diff(f2,y)-diff(f1,y)diff(f2,x),xyt-1; ideal j=std(i); ideal k =eliminate(j, xy*t); k; k[1]=48c12+1280c11+12288c10+49152c9+65467c8-2560c7-24576c6- 98304c5-131078c4+1280c3+12288c2+49152c+65563 *

The roots of this last polynomial in $c$ correspond to systems with a degenerate solution, and we can check that the only positive root of $f_{1},f_{2}$ and their jacobian is $1$. We check, again using Singular [8], with the library “signcond.lib” (implemented by E. Tobis, based on the algorithms described in [1]) that if we take for example $c=\frac{1}{2}$ ($c<1$), the system has 3 positive solutions, and if we take $c=\frac{8}{7}$ ($c>1$), the system has only $1$ positive solution. We use the command firstoct, that computes the number of roots of a system in the first octant, that is, the positive roots.

LIB "signcond.lib"; ring r=0, (x,y), dp; poly f1=-y^4-x^5y^4+x^2y^8+x^3; poly f2=-(3*(1/2)+8)y^4-(1/2)x^5y^4+(2(1/2)+8)x^2y^8+2x^3y^5; poly f3=-(3*(8/7)+8)y^4-(8/7)x^5y^4+(2(8/7)+8)x^2y^8+2x^3y^5; ideal i2 = f1,f2; ideal j2 = std(i2); firstoct(j2); 3 ideal i3 = f1,f3; ideal j3 = std(i3); firstoct(j3); 1 *

For $c=\frac{1}{2}$, the condition in [13] to ensure exactly one positive solution is trivially not satisfied (as expected since the system has $3$ positive solutions).

Observe that this procedure is symbolic and thus certified. This computation of the number of positive solutions with the command firstoct works for moderately sized polynomial systems with coefficients in $\mathbb{Q}$ or an algebraic extension of it. Our results are particularly useful to study families of polynomials.

4. Dominating matrices

In this section, we present some conditions on $A$ and $C$ that guarantee that the hypotheses of Theorem 3.8 are satisfied. Our main result is Theorem 4.6.

We first present conditions that guarantee that a matrix $A$ admits a choice of a Gale dual matrix $B$, which satisfies condition (1) of Theorem 3.8, for any uniform matrix of coefficients $C$ satisfying (2.3) (which means that it does not depend on $I_{C}$). When $A\in\mathbb{Z}^{(d+1)\times n}$ we will relate these conditions with complete intersection lattice ideals in Section 6.

We recall some definitions from [11], with the difference that we replace rows by columns and allow matrices with real entries.

Definition 4.1**.**

A vector is said to be mixed if contains a strictly positive and a strictly negative coordinate. More generally, a real matrix is called mixed if every column contains a strictly positive and a strictly negative entry. A real matrix is called dominating if it contains no square mixed submatrix. An empty matrix is mixed and also dominating.

For example any matrix with the following sign pattern

[TABLE]

is mixed and dominating. When a matrix is mixed and dominating we will say that it is mixed dominating.

Observe that since a matrix $A$ as in (2.1) has a row of ones, the columns of any Gale dual matrix $B$ add up to zero, and thus $B$ is always mixed. Also note that a mixed matrix is weakly mixed (see Definition 3.7), but the converse is not true in general as a weakly mixed matrix can also contain a column with only zero entries.

We now show that for uniform matrices $A$ admitting a dominating Gale dual matrix $B$, condition (1) in Theorem 3.8 is automatically satisfied for any uniform matrix $C$ having a positive vector in the kernel.

Lemma 4.2**.**

Assume that $A\in\mathbb{R}^{(d+1)\times n}$ is a uniform matrix. If $B\in\mathbb{R}^{n\times k}$ is a Gale dual matrix of $A$ which is dominating, then condition (1) of Theorem 3.8 is satisfied for all $C\in\mathbb{R}^{d\times n}$ uniform satisfying ${\bf 0}\in\mathcal{C}^{\circ}$.

Proof.

Let $C\in\mathbb{R}^{d\times n}$ uniform, and take any Gale dual matrix $D\in\mathbb{R}^{n\times(k+1)}$ of $C$ such that $\Delta_{P}$ is nonempty and bounded (which exists due to Lemma 2.3 and Corollary 2.2).

If $L=\{\ell\}\in\mathcal{F}(\Delta_{P})$, then $B_{L}$ weakly mixed means that it has only zeros, which forces the matrix $A$ minus the $\ell$-th column to have rank $<d+1$. Consider $L\in\mathcal{F}(\Delta_{P})$ such that $|L|\geq 2$. Note that $|L|\leq k$ since $C$ is uniform. If $B_{L}$ is weakly mixed then at least $k-|L|+1$ columns of $B_{L}$ contain only zero entries, for otherwise we would get a square submatrix of size $|L|\times|L|$ containing a positive and a negative coefficient in each column. But if $k-|L|+1$ columns of $B_{L}$ contain only zeros, then we get $k-|L|+1$ linearly independent vectors in the kernel of the matrix $A_{\setminus L}$ obtained from $A$ by removing the columns indexed by $L$. This forces the matrix $A_{\setminus L}$ to have rank less than $d+1$, which is a contradiction, since $A$ is uniform. ∎

The following results will be useful. These propositions are only stated for matrices with integer coefficients in [11], but clearly the proofs given in that paper also work for real matrices.

Proposition 4.3** ([11], Corollary 2.7 and 2.8).**

If a real matrix is mixed dominating, then any nonzero linear combination of its columns is a mixed vector. In particular, its columns are linearly independent.

The proof of this proposition is a consequence of Proposition 2.6 in [11], which implies that the set of indices for the positive entries of any nonzero linear combination of the columns of a dominating matrix contains the set of indices of the positive or negative entries of some column of the matrix. Moreover, they show that the following is true.

Proposition 4.4** ([10], Proposition 4.1).**

The left kernel of any mixed dominating real matrix contains a positive vector.

We will also need the following Lemma.

Lemma 4.5**.**

Assume that $C\in\mathbb{R}^{d\times n}$ has maximal rank $d$ and that ${\bf 0}\in\mathcal{C}^{\circ}$. Let $\tilde{D}\in\mathbb{R}^{n\times k}$ be any matrix of maximal rank $k$ such that $C\tilde{D}=0$. Assume that

[TABLE]

where $\tilde{P}_{1},\ldots,\tilde{P}_{n}$ are the row vectors of $\tilde{D}$. Then, there exists a positive vector $D_{0}$ in the kernel of $C$ which does not belong to the linear span of the column vectors of $\tilde{D}$, and the matrix $D\in\mathbb{R}^{n\times(k+1)}$ obtained from $\tilde{D}$ by adding $D_{0}$ as a first column vector is Gale dual to $C$ and satisfies $(1,0,\ldots,0)\in{\mathcal{C}}_{P}$.

Proof.

By (4.1) there exists a positive vector in the left kernel of $\tilde{D}$, in other words, a row vector $\lambda$ with positive coordinates 222In fact, it is sufficient that $\lambda$ is a nonzero vector with only nonnegative coordinates. such that $\lambda\cdot\tilde{D}=(0,\ldots,0)$. Since ${\bf 0}\in\mathcal{C}^{\circ}$ we have $\ker(C)\cap\mathbb{R}_{>0}^{n}\neq\emptyset$. Then, as $\ker(C)$ has dimension $k+1$ and $\tilde{D}$ has rank $k$, there exists a vector $D_{0}\in\ker(C)\cap\mathbb{R}_{>0}^{n}$ which does not belong to the linear span of the column vectors of $\tilde{D}$. The matrix $D\in\mathbb{R}^{n\times k}$ obtained from $\tilde{D}$ by adding $D_{0}$ as a first column vector is Gale dual to $C$. Moreover, we have $\lambda\cdot D=(\lambda\cdot D_{0},0,\ldots,0)$ and thus $(1,0,\ldots,0)\in{\mathcal{C}}_{P}$ since $\lambda\cdot D_{0}>0$ (here $\lambda$ is a row vector, $D_{0}$ is a column vector so that $\lambda\cdot D_{0}$ is a real number, which is positive since $\lambda$ and $D_{0}$ are positive vectors). ∎

If $S\subset\mathbb{R}^{n}$ if a subspace, we denote $\operatorname{sign}(S)=\{\operatorname{sign}(v)\,:\,v\in S\}$. Recall that we denote the column vectors of a matrix $B$ by $B_{1},\dots,B_{k}$.

Theorem 4.6**.**

Let $\mathcal{A}=\{a_{1},\dots,a_{n}\}\subset\mathbb{R}^{d}$. Assume $A\in\mathbb{R}^{(d+1)\times n}$ as in (2.1) and $C\in\mathbb{R}^{d\times n}$ are uniform matrices. Suppose there exist a dominating Gale dual matrix $B\in\mathbb{R}^{n\times k}$ of $A$. Assume ${\bf 0}\in{\mathcal{C}}^{\circ}$ and $\operatorname{sign}(B_{j})\in\operatorname{sign}(\ker(C))$ for each $j=1,\dots,k$. Then, $n_{\mathcal{A}}(C)>0$.

Proof.

As $B$ is dominating and $A,C$ are uniform, condition (1) of Theorem 3.8 is satisfied by Lemma 4.2. As $\operatorname{sign}(B_{j})\in\operatorname{sign}(\ker(C))$ for $j=1,\dots,k$, there exist vectors $D_{1},\dots,D_{k}$ in $\ker(C)$ such that $\operatorname{sign}(D_{j})=\operatorname{sign}(B_{j})$ for each $j=1,\dots,k$. Consider the matrix $\tilde{D}$ with column vectors $D_{1},\ldots,D_{k}$. Since $B$ is mixed dominating (it is mixed since $A$ contains a row of ones) the matrix $\tilde{D}$ is mixed dominating and furthermore $\tilde{D}$ has rank $k$ by Proposition  4.3. Moreover, by Proposition 4.4, there is a positive vector in the left kernel of $\tilde{D}$. Then, condition (4.1) is satisfied, and thus by Lemma 4.5 and Corollary 2.2, there is a positive vector $D_{0}$ such that the matrix $D$ with column vectors $D_{0},\ldots,D_{k}$ is Gale dual to $C$ and the associated polytope $\Delta_{P}$ is nonempty and bounded. By construction, condition (2) of Theorem 3.8 is also satisfied, and thus $n_{\mathcal{A}}(C)>0$. ∎

Recall that the support of a vector $v\in\mathbb{R}^{n}$ is defined to be the set of its nonzero coordinates, and we denote it by $\operatorname{supp}(v)$. Given a subspace $S\subset\mathbb{R}^{n}$, a circuit of $S$ is a nonzero element $s\in S$ with minimal support (with respect to inclusion). Given a vector $v$, a circuit $s=(s_{1},\dots,s_{n})$ is said to be conformal to $v=(v_{1},\dots,v_{n})$ if for any index $i$ in $\operatorname{supp}(s)$, $\operatorname{sign}(s_{i})=\operatorname{sign}(v_{i})$. The next lemma shows that if $A$ admits a Gale dual mixed dominating matrix, then there exist a choice of Gale mixed dominating matrix of $A$ whose columns are circuits of $\ker(A)$. Note that all the circuits of $\ker(A)$ can be described in terms of vectors of maximal minors of $A$, and so they only depend on the associated oriented matroid of $A$.

Lemma 4.7**.**

Assume $A\in\mathbb{R}^{(d+1)\times n}$ as in (2.1). Suppose there exist a dominating Gale dual matrix $B\in\mathbb{R}^{n\times k}$ of $A$. Then, there exists a dominating Gale dual matrix $B^{\prime}\in\mathbb{R}^{n\times k}$ of $A$ such that every column of $B^{\prime}$ is a circuit of $\ker(A)$.

Proof.

It is a known result that every vector in $\ker(A)$ can be written as a nonnegative sum of circuits conformal to it (see  [17]). In particular, for every vector in $\ker(A)$, there exists a circuit conformal to it. For each column $B_{i}$ of $B$, $i=1,\dots,k$, take a circuit $B^{\prime}_{i}$ of $\ker(A)$ such that $B^{\prime}_{i}$ is conformal to $B_{i}$. Now, we take $B^{\prime}$ the matrix with columns $B^{\prime}_{1},\dots,B^{\prime}_{k}$. Every column of $B^{\prime}$ is a circuit of $\ker(A)$, $B^{\prime}$ is mixed since $A$ has a row of ones, and is dominating because $B^{\prime}_{i}$ is conformal to $B_{i}$ for each $i=1,\dots,k$ and the matrix $B$ is dominating. Since $B^{\prime}$ is mixed dominating, the columns of $B^{\prime}$ are linearly independent by Proposition 4.3, and then $B^{\prime}$ is a Gale dual matrix of $A$. ∎

5. Geometric conditions on $A$ and $C$

The main result of this section is Theorem 5.8, where we give geometric conditions on $A$ and $C$ that guarantee that the hypotheses of Theorem 3.8 are satisfied.

A characterization of matrices $A$ admitting a mixed dominating Gale dual matrix $B$ can be found in [10]. Recall that our definition of mixed dominating matrix differs from the one in [10] by replacing rows by columns. Here we present this result with our notation. We denote the convex hull of a point configuration $\mathcal{A}$ by $\operatorname{{\rm chull}}(\mathcal{A})$. Recall also that we assume $n\geq d+2$, so that $\mathcal{A}$ cannot be the set of vertices of a $d$-dimensional simplex.

Theorem 5.1** ([10], Theorem 5.6 ).**

Let $\mathcal{A}=\{a_{1},\dots,a_{n}\}\subset\mathbb{R}^{d}$, with $n\geq d+2$ and $A\in\mathbb{R}^{(d+1)\times n}$ as in (2.1). Then $A$ admits a mixed dominating Gale dual matrix $B$ if and only if $\mathcal{A}$ can be written as a disjoint union $\mathcal{A}={\mathcal{A}}_{1}\sqcup{\mathcal{A}}_{2}$ such that

1. (1)

the polytopes $\operatorname{{\rm chull}}({\mathcal{A}}_{1})$ and $\operatorname{{\rm chull}}({\mathcal{A}}_{2})$ intersect in exactly one point, 2. (2)

the corresponding matrices $A_{1}$ and $A_{2}$ as in (2.1) of $\mathcal{A}_{1}$ and $\mathcal{A}_{2}$ respectively, admit mixed dominating Gale dual matrices, and 3. (3)

$\dim\operatorname{{\rm chull}}({\mathcal{A}})=\dim\operatorname{{\rm chull}}({\mathcal{A}}_{1})+\dim\operatorname{{\rm chull}}({\mathcal{A}}_{2})$.

Moreover, we have:

Lemma 5.2** ([10], Corollary 5.7).**

If $A$ admits a mixed dominating Gale dual matrix $B$ then $\operatorname{{\rm chull}}({\mathcal{A}})$ has at most $2d$ vertices.

In particular, by Lemma 4.2, we have:

Corollary 5.3**.**

Let $\mathcal{A}=\{a_{1},\dots,a_{n}\}\subset\mathbb{R}^{d}$. Assume that $A$ as in (2.1) is uniform and that $\mathcal{A}\subset\mathbb{R}^{d}$ can be decomposed as a disjoint union $\mathcal{A}={\mathcal{A}}_{1}\sqcup{\mathcal{A}}_{2}$ such that conditions (1), (2) and (3) of Theorem 5.1 hold. Then, there exists a Gale dual matrix $B\in\mathbb{Z}^{n\times k}$ of $A$ such that condition (1) of Theorem 3.8 is satisfied.

The following observation says that if we have a point configuration $\mathcal{A}_{v}\subset\mathbb{R}^{d}$ such that the corresponding matrix $A_{v}$ admits a Gale dual mixed dominating matrix, then, for any other point configuration $\mathcal{A}\subset\mathbb{R}^{d}$ that contains $\mathcal{A}_{v}$ and their convex hulls $\operatorname{{\rm chull}}(\mathcal{A}),\operatorname{{\rm chull}}(\mathcal{A}_{v})$ coincide (that is, $\mathcal{A}$ can be obtained from $\mathcal{A}_{v}$ adding points inside the convex hull), the corresponding matrix $A$ also admits a Gale dual mixed dominating matrix.

Lemma 5.4**.**

Let $\mathcal{A}=\{a_{1},\dots,a_{n}\}$, $\mathcal{A}_{v}\subset\mathbb{R}^{d}$ be two point configurations such that $\mathcal{A}_{v}\subset\mathcal{A}$. Assume that the corresponding matrix $A\in\mathbb{R}^{(d+1)\times n}$ is uniform and that the following conditions hold:

1. (1)

$\operatorname{{\rm chull}}(\mathcal{A})=\operatorname{{\rm chull}}(\mathcal{A}_{v})$** 2. (2)

The corresponding matrix $A_{v}\in\mathbb{R}^{(d+1)\times|A_{v}|}$ has a Gale dual matrix $B_{v}$ which is dominating.

Then, there exists a a mixed dominating Gale dual matrix $B\in\mathbb{Z}^{n\times k}$ of $A$ and thus condition (1) of Theorem 3.8 is satisfied.

Note that Lemma 5.4 follows from applying several times Theorem 5.1 (taking one point from ${\mathcal{A}}_{v}$ as ${\mathcal{A}}_{2}$), but we present a constructive proof.

Proof of Lemma 5.4.

Without loss of generality, we may assume that $\mathcal{A}_{v}=\{a_{1},\ldots,a_{s}\}$, with $s\geq d$. For $i=s+1,\ldots,n$, there exists a subset $\mathcal{A}_{i}$ of $\mathcal{A}_{v}$ such that $\mathcal{A}_{i}$ is the set of vertices of a $d$-simplex and $a_{i}$ is contained in the interior of $\operatorname{{\rm chull}}(\mathcal{A}_{i})$. Then there exists an affine relation on $\mathcal{A}_{i}\cup\{a_{i}\}$ where the coefficient of $a_{i}$ is equal to one and the coefficients of the points of $\mathcal{A}_{i}$ are all negative. Using the affine relations on $\mathcal{A}_{v}$ given by the column vectors of $B_{v}$, we get $k$ linearly independent vectors in the kernel of $A$ which are the column vectors of a upper triangular block matrix $B$ Gale dual to $A$ of the following form:

[TABLE]

where $R$ has only nonpositive entries (and at least two negative entries in each column) and $I_{n-s}$ is the identity matrix of size $n-s$. Clearly, if $B_{v}$ is dominating then $B$ is dominating and thus by Lemma 4.2 the first item of Theorem 3.8 is satisfied. ∎

We have the following corollary.

Corollary 5.5**.**

Let $\mathcal{A}=\{a_{1},\dots,a_{n}\}$, $\mathcal{A}_{v}\subset\mathbb{R}^{d}$ be two point configurations such that $\mathcal{A}_{v}\subset\mathcal{A}$ and $\operatorname{{\rm chull}}(\mathcal{A})=\operatorname{{\rm chull}}(\mathcal{A}_{v})$. Assume that the corresponding matrix $A$ is uniform and that $\mathcal{A}_{v}$ is either the set of vertices of $d$-simplex, or a circuit in $\mathbb{R}^{d}$. Then, there exists a Gale dual matrix $B\in\mathbb{Z}^{n\times k}$ of $A$ such that condition (1) of Theorem 3.8 is satisfied.

Consider $\mathcal{A}$ and the point configuration $\mathscr{C}=\{C_{1},\dots,C_{n}\}$ given by the columns of the coefficient matrix $C$. Consider the $(d+1)\times n$-matrix

[TABLE]

and assume that $A$ and $\bar{C}$ are uniform. Given a subset $J\subset\{1,\dots,n\}$, we denote $\mathcal{A}_{J}=\{a_{j}\,:\,j\in J\}$.

Definition 5.6**.**

Given a subset $I\subset\{1,\dots,n\}$, we say that $A$ and $C$ are $I$-compatible if the following conditions hold:

1. (1)

The corresponding matrices $A_{I}$ and $\overline{C}_{I}$ admit Gale dual matrices which are mixed dominating and have the same sign pattern, 2. (2)

$\operatorname{{\rm chull}}(\mathcal{A}_{I})=\operatorname{{\rm chull}}(\mathcal{A})$* and $\operatorname{{\rm chull}}(\mathscr{C}_{I})=\operatorname{{\rm chull}}(\mathscr{C})$,* 3. (3)

*For each $j\notin I$, there exist $J\subset I$, with $|J|=d+1$, such that $a_{j}\in\operatorname{{\rm chull}}(\mathcal{A}_{J})$ and $C_{j}\in\operatorname{{\rm chull}}({\mathscr{C}}_{J})$. *

The condition that $A$ and $C$ are $I$-compatible can be translated in terms of signs of maximal minors of $A$ and $\bar{C}$. Also note that the configurations $\mathcal{A}$ and $\mathscr{C}$ may have different oriented matroids. The following Example 5.7 shows two $I$-compatible configurations with different oriented matroids.

Example 5.7**.**

We show in Figure 2 an example of two point configurations, $\mathcal{A}=\{a_{1},\dots,a_{6}\}$ and ${\mathscr{C}}=\{C_{1},\dots,C_{6}\}$ with $d=2$ and $k=3$, which are $I$-compatible, for $I=\{1,2,3,4\}$. In this case $a_{5}\in ch(\mathcal{A}_{I_{5}})$, $C_{5}\in ch({\mathscr{C}}_{I_{5}})$ for $I_{5}=\{1,3,4\}$ and $a_{6}\in ch(\mathcal{A}_{I_{6}})$, $C_{6}\in ch({\mathscr{C}}_{I_{6}})$ for $I_{6}=\{1,2,3\}$.

We have the following result:

Theorem 5.8**.**

Assume that $A$, $C$ and $\bar{C}$ are uniform. Suppose ${\bf 0}\in{\mathcal{C}}^{\circ}$, and there exists $I\subset\{1,\dots,n\}$ such that $A$ and $C$ are $I$-compatible. Then, $n_{\mathcal{A}}(C)>0$.

Proof.

Let $B_{I}$ be a Gale dual matrix of $A_{I}$ as in Condition (1) of Definition 5.6. As $a_{j}\in\operatorname{{\rm chull}}(\mathcal{A}_{I})$ for each $j\notin I$, we can use Lemma 5.4. We construct a dominating matrix $B$, using the matrix $B_{I}$ and using for each $a_{j}$, $j\notin I$, the affine relation given by the circuit $a_{j}\cup\mathcal{A}_{J}$, with $J$ as in Condition (3) of Definition 5.6, to obtain a vector in the kernel of $A$ as in the proof of Lemma 5.4. Conditions (1) and (3) of the definition of being $I$-compatible mean that there exist $k$ vectors in the kernel of $\bar{C}$ with the same sign patterns as the columns of the constructed $B$, and these $k$ vectors are linearly independent because they form a mixed dominating matrix (Proposition 4.3). We have that $\ker{(\bar{C})}\subset\ker(C)$, and $\operatorname{sign}(B_{1}),\dots,\operatorname{sign}(B_{k})\in\operatorname{sign}(\ker(C))$. We can apply Theorem 4.6 and then $n_{\mathcal{A}}(C)>0$. ∎

Remark 5.9**.**

When $|I|=d+2$, condition (1) in Definition 5.6 can be translated in terms of signatures of circuits. Given a circuit $\mathcal{U}=\{u_{1},\dots,u_{d+2}\}\subset\mathbb{R}^{d}$, and a nonzero affine relation $\lambda\in\mathbb{R}^{d+2}$ among the $u_{i}$, we call $\Lambda_{+}=\{i\in\{1,\dots,d+2\}\,:\,\lambda_{i}>0\}$ and $\Lambda_{-}=\{i\in\{1,\dots,d+2\}\,:\,\lambda_{i}<0\}$. The pair $(\Lambda_{+},\Lambda_{-})$ is usually called a signature of $\mathcal{U}$. As $\mathcal{U}$ is a circuit, the pairs $(\Lambda_{+},\Lambda_{-})$ and $(\Lambda_{-},\Lambda_{+})$ are the two possible signatures. Then, we consider the (unordered) signature partition $\mathcal{S}(\mathcal{U})=\{\Lambda_{+},\Lambda_{-}\}$. Given a subset $I\subset\{1,\dots,n\}$, with $|I|=d+2$, and $A$ and $C$ uniform, condition (1) in Definition 5.6 is equivalent to the following condition:

1. (1’)

$\mathcal{S}(\mathcal{A}_{I})=\mathcal{S}({\mathscr{C}}_{I})$**

In this case, condition (3) in Definition 5.6 implies that $\{a_{j}\}\cup\mathcal{A}_{J}$ and $\{C_{j}\}\cup{\mathscr{C}}_{J}$ have the same signature partition, which is $(d+1,1)$.

5.1. The case $k=2$

The point configurations such that the corresponding matrix admits a Gale dual which is dominating are limited. So, if we are not in this case, checking condition (1) of Theorem 3.8 involves knowing the incidences of the facets of the polytope $\Delta_{P}$. However, we now show that in case $A$ has codimension $k=2$, there always exists a choice of Gale dual matrix $B$ such that we can conclude that $n_{\mathcal{A}}(C)>0$ with the help of Theorem 3.8 without checking Condition (1) as it becomes a consequence of the other conditions.

Lemma 5.10**.**

Assume that $A$ and $C$ are uniform matrices and $k=2$. Suppose that ${\bf 0}\in\mathcal{C}^{\circ}$. Then there exists a matrix $B$ Gale dual to $A$ such that for any matrix $D$ Gale dual to $C$ for which $\Delta_{P}$ is nonempty, bounded and the condition (2) of Theorem 3.8 is satisfied, condition (1) of Theorem 3.8 is satisfied, and thus $n_{\mathcal{A}}(C)>0$.

Proof.

Let $B$ be any Gale dual matrix of $A$ with row vectors $b_{1},\ldots,b_{{n}}$. Choose any $i_{2}\in I_{C}$. Then there exist $i_{1}\in I_{C}$ such that the cone $\mathbb{R}_{>0}b_{i_{1}}+\mathbb{R}_{<0}b_{i_{2}}$ does not contain vectors $b_{i}$ with $i\in I_{C}$. Note that the latter cone has dimension two since $A$ is uniform (which implies that $B$ is uniform as well). There exists a matrix $R$ of rank two such that $B_{\{i_{1},i_{2}\}}\cdot R=\operatorname{I}_{2}$ (if we assume that $A$, $B$ have integer entries, then there exists an integer matrix $R$ of rank two such that $B_{\{i_{1},i_{2}\}}\cdot R=a\cdot\operatorname{I}_{2}$ where $a=|\det(B_{\{i,j\}})|$). Consider the matrix $B^{\prime}=B\cdot R$, with row vectors $b^{\prime}_{1},\ldots,b_{n}^{\prime}$. Then $B^{\prime}$ is a Gale dual matrix to $A$ such that $b^{\prime}_{i_{1}}=(1,0)$, $b^{\prime}_{i_{2}}=(0,1)$ and the open quadrant $\mathbb{R}_{>0}\times\mathbb{R}_{<0}$ does not contain any vector $b_{i}^{\prime}$ with $i\in I_{C}$. Note also that if $i\in I_{C}$ and $i\neq i_{1},i_{2}$ then both coordinates of $b_{i}^{\prime}$ are nonzero for otherwise this would give a vanishing maximal minor of $B^{\prime}$. In particular, we get $b_{i}^{\prime}\neq 0$, and thus $b_{i}^{\prime}$ is not weakly mixed, for all $i\in I_{C}$. Suppose now that there are two distinct vectors $b_{i}^{\prime}$ and $b_{j}^{\prime}$ with $i,j\in I_{C}$ such that the submatrix $B_{\{i,j\}}$ is weakly mixed. Then these row vectors lie in opposite quadrants of $\mathbb{R}^{2}$ and these quadrants should be $\mathbb{R}_{>0}^{2}$ and $\mathbb{R}_{<0}^{2}$. But then the cone $\mathbb{R}_{>0}b_{i}^{\prime}+\mathbb{R}_{>0}b_{j}^{\prime}$ contains either $b^{\prime}_{i_{1}}=(1,0)$ or $b^{\prime}_{i_{2}}=(0,1)$, and thus $\{i,j\}\notin F_{L}$. ∎

Given a vector $v\in\mathbb{R}^{n}$ and $I\subset\{1,\dots,n\}$ we denote by $v_{I}\in\mathbb{R}^{|I|}$ the vector obtained from $v$ after removing the coordinates with indexes that do not belong to $I$. Given a set $S\subset\mathbb{R}^{n}$, we denote $S_{I}=\{v_{I}\,:\,v\in S\}$.

Consider the four open quadrants of $\mathbb{R}^{2}$ numbered from $1$ to $4$, where the signs of the two coordinates are $(+,+)$, $(-,+)$, $(-,-)$, and $(+,-)$ for the first, second, third and fourth quadrant respectively. In case that there exists a Gale dual matrix $B$ with rows in each of the quadrants, we have the following result.

Lemma 5.11**.**

Given $A\in\mathbb{R}^{(d+1)\times(d+3)}$ uniform, let $B\in\mathbb{R}^{(d+3)\times 2}$ be a Gale dual matrix of $A$. Suppose there exists rows of $B$, $b_{i_{j}}$, with $1\leq j\leq 4$, such that $b_{i_{j}}$ lies in the $j$-th open quadrant of $\mathbb{R}^{2}$. Let $C\in\mathbb{R}^{d\times n}$ uniform. Suppose that ${\bf 0}\in{\mathcal{C}}^{\circ}$. Assume moreover that given a Gale dual matrix of $C$, the row vectors $P_{i_{1}},\dots,P_{i_{4}}$ define normals to facets of the closure of the cone ${\mathcal{C}}_{P}^{\nu}$ in (2.5). If $\operatorname{sign}((B_{j})_{I_{C}})\in\operatorname{sign}((\ker(C))_{I_{C}})$ for $j=1,2$, then $n_{\mathcal{A}}(C)>0$.

Note that the condition that $P_{i_{1}},\dots,P_{i_{4}}$ define normals to facets of the associated cone ${\mathcal{C}}_{P}^{\nu}$ is independent of the choice of Gale dual matrix of $C$.

Proof.

As $\operatorname{sign}((B_{j})_{I_{C}})\in\operatorname{sign}((\ker(C))_{I_{C}})$ for $j=1,2$, there are vectors $D_{1},D_{2}\in\ker(C)$ such that $\operatorname{sign}((D_{j})_{I_{C}})=\operatorname{sign}((B_{j})_{I_{C}})$ for each $j=1,2$. We can assume that $D_{1}$ and $D_{2}$ are linearly independent. If not, the zero coordinates of $D_{1}$ and $D_{2}$ (which are at most two, since $C$ is uniform) have to be the same. That is, $(D_{1})_{j}=0$ if and only if $(D_{2})_{j}=0$ (otherwise, they cannot be linearly dependent). Suppose that $(D_{1})_{j}=(D_{2})_{j}=0$ for certain $j$. If $j\in I_{C}$, then $(B_{1})_{j}=(B_{2})_{j}=0$, but since $A$ is uniform, $B_{1}$ and $B_{2}$ have at most one zero coordinate, and then, $B_{1}$ and $B_{2}$ are scalar multiples of each other, a contradiction. Then if $(D_{1})_{j}=(D_{2})_{j}=0$, $j\notin I_{C}$. We take a vector $v$ in $\ker(C)$ such that $D_{1}$ and $v$ are linearly independent. Then we can take $D^{\prime}_{2}=D_{2}+\lambda v$, with $\lambda$ small enough such that $\operatorname{sign}((D_{2})_{I_{C}})=\operatorname{sign}((B_{2})_{I_{C}})$.

So, we can suppose that $D_{1}$ and $D_{2}$ are linearly independent. Consider the matrix $\tilde{D}$ with column vectors $D_{1}$ and $D_{2}$. We have that ${\bf 0}$ belongs to the open cone generated by the rows of $\tilde{D}$, because the $i_{j}$-th row of $\tilde{D}$ belongs to the $j$-th open quadrant, then Condition 4.1 of Lemma 4.5 is satisfied. As ${\bf 0}\in{\mathcal{C}}^{\circ}$, by Lemma 4.5 and Corollary 2.1, there exists a positive vector $D_{0}$ such that the matrix obtained from $\tilde{D}$ by adding $D_{0}$ as a first column vector is Gale dual to $C$ and the associated polytope $\Delta_{P}$ is nonempty and bounded. Also note that $\Delta_{P}$ has a facet for each row vector $i_{j}$ of $\tilde{D}$, each one in the $j$-quadrant of $\mathbb{R}^{2}$, for $j=1,\dots,4$. Then, if we have a $2\times 2$ mixed submatrix of $B$, it does not correspond to a submatrix $B_{L}$, with $L\in\mathcal{F}(\Delta_{P})$ (and any row of $\tilde{D}$ corresponding to $i\in I_{C}$ is not equal to zero). Then, all the conditions of Theorem 3.8 are satisfied and $n_{\mathcal{A}}(C)>0$.

∎

6. Algebraic conditions and real solutions of integer configurations

In this section we will consider integer configurations $\mathcal{A}$ and thus, integer matrices $A$. Interestingly, in Corollary 6.2 we will relate Lemma 4.2 with known algebraic results in the study of toric ideals [21, Ch.4]. Indeed, we summarize in § 6.1 some known algebraic results that show the existence of a mixed dominating Gale dual matrix is equivalent to the fact that there is a full dimensional sublattice of the integer kernel ${\rm ker}_{\mathbb{Z}}(A)$ whose associated lattice ideal (6.1) is a complete intersection. This means that it can be generated by as many polynomials as the codimension of its zero set. In the opposite spectrum, when an ideal is not Cohen-Macaulay there is no such direct relation between algebra and geometry (see for instance [9], or Chapter 1 in [20] in the graded case). When $k=2$, we also consider lattice ideals which are not Cohen-Macaulay. Proposition 6.4 shows how to deal with this more complicated algebraic case. Also, in § 6.2 we naturally extend the search for positive solutions to the search for real solutions with nonzero coordinates.

6.1. Algebraic conditions

A polynomial ideal is called binomial if it can be generated with polynomials with at most two terms. A subgroup $\mathcal{L}\subset\mathbb{Z}^{n}$ is called a lattice. We associate to a lattice $\mathcal{L}$ the following binomial ideal:

[TABLE]

where $u=u^{+}-{u}^{-}$ is the decomposition in positive and negative components. For example, if $u=(1,-2,1,0)\in\mathbb{Z}^{4}$, then $x^{u^{+}}-x^{u^{-}}=x_{1}x_{3}-x_{2}^{2}$.

Given a configuration $\mathcal{A}=\{a_{1},\dots,a_{n}\}\subset\mathbb{Z}^{d}$ of integral points, and the associated matrix $A\in\mathbb{Z}^{(d+1)\times n}$, let $B\in\mathbb{Z}^{n\times k}$ be a Gale dual matrix of $A$, and denote by $B_{1},\dots,B_{k}$ the column vectors of $B$. Note that $\{B_{1},\dots,B_{k}\}$ is a $\mathbb{Q}$-basis of $\ker_{\mathbb{Z}}(A)$, but it is not necessarily a $\mathbb{Z}$-basis unless the greatest common divisor of the maximal minors of $B$ is equal to $1$. When this is the case, we will say that $B$ is a $\mathbb{Z}$-Gale dual of $A$. We associate to any choice of Gale dual $B$ of $A$ the following lattice:

[TABLE]

and its corresponding lattice ideal $I_{\mathcal{L_{B}}}$. In particular, when $\mathcal{L}_{B}=\ker_{\mathbb{Z}}(A)$, then the lattice ideal $I_{\mathcal{L}_{B}}$ is known as the toric ideal $I_{A}$. We have the following known result from [11]. See also Theorem 2.1 of [12], where the notation is similar to the notation of this paper.

Theorem 6.1** ([11], Theorem 2.9).**

The lattice ideal $I_{\mathcal{L}_{B}}$ is a complete intersection if and only if $\mathcal{L}_{B}=\mathcal{L}_{B^{\prime}}$ for some dominating matrix $B^{\prime}\in\mathbb{Z}^{n\times k}$. In this case, $I_{\mathcal{L}_{B}}=\langle x^{u^{+}}-x^{u^{-}}\,:\,u\text{ is a column of }B^{\prime}\rangle$.

The following result is a direct consequence of Lemma 4.2 and Theorem 6.1.

Corollary 6.2**.**

If $A\in\mathbb{Z}^{(d+1)\times n}$ and $C\in\mathbb{R}^{d\times n}$ are uniform matrices and $B\in\mathbb{Z}^{n\times k}$ is a Gale dual matrix of $A$ such that the lattice ideal $I_{\mathcal{L}_{B}}$ is a complete intersection, then there exists a Gale dual matrix $B^{\prime}\in\mathbb{Z}^{n\times k}$ of $A$ which satisfies the condition (1) of Theorem 3.8.

Given $A$, let $B\in\mathbb{Z}^{n\times k}$ a Gale dual matrix of $A$, and consider the lattice $\mathcal{L}_{B}=\mathbb{Z}B$. The set of rows of $B$, $\{b_{1},\dots,b_{n}\}\subset\mathbb{Z}^{k}$ is called a Gale diagram of $\mathcal{L}_{B}$. Any other $\mathbb{Z}$-basis for $\mathcal{L}_{B}$ yields a Gale diagram, which means that Gale diagrams are unique up to transformation by an invertible integer matrix.

The following proposition from [16] relates Gale diagrams with algebraic properties of the lattice ideal $\mathcal{L}_{B}$ when $k=2$:

Proposition 6.3** ([16], Proposition 4.1).**

Given $A\in\mathbb{Z}^{(d+1)\times(d+3)}$, let $B\in\mathbb{Z}^{n\times 2}$ be a Gale dual matrix of $A$. The lattice ideal $I_{\mathcal{L}_{B}}$ is not Cohen-Macaulay if and only if it has a Gale diagram which intersects all the four open quadrants of $\mathbb{R}^{2}$.

The following result follows from Proposition 6.3 and Lemma 5.11.

Proposition 6.4**.**

Given $A\in\mathbb{Z}^{(d+1)\times(d+3)}$ uniform, let $B\in\mathbb{Z}^{n\times 2}$ be a Gale dual matrix of $A$. Suppose that the lattice ideal $I_{\mathcal{L}_{B}}$ is not Cohen-Macaulay and let $B^{\prime}$ be any other Gale Dual matrix of $A$ such that the columns $B^{\prime}_{1},B^{\prime}_{2}$ of $B^{\prime}$ form a $\mathbb{Z}$-basis of $\mathcal{L}_{B}$ and such that the corresponding Gale diagram $\{b^{\prime}_{1},\dots,b^{\prime}_{n}\}$ intersects all the four open quadrants of $\mathbb{R}^{2}$. Let $b^{\prime}_{i_{j}}$, with $1\leq j\leq 4$, be rows of $B^{\prime}$ each lying in the interior of a different open quadrant in $\mathbb{R}^{2}$. Let $C\in\mathbb{R}^{d\times n}$ uniform satisfying ${\bf 0}\in{\mathcal{C}}^{\circ}$. Assume moreover that given a Gale dual matrix of $C$, the row vectors $P_{i_{1}},\dots,P_{i_{4}}$ define normals to facets of the closure of the cone ${\mathcal{C}}_{P}^{\nu}$ in (2.5).

Then, if $\operatorname{sign}((B^{\prime}_{j})_{I_{C}})\in\operatorname{sign}((\ker(C))_{I_{C}})$ for $j=1,2$, then $n_{\mathcal{A}}(C)>0$.

6.2. Real solutions

When $A$ has integer entries, (1.1) is a system of Laurent polynomials with real coefficients, which are defined over the real torus $(\mathbb{R}^{*})^{d}$. In this subsection, we are interested on the existence of real solutions of  (1.1) with nonzero coordinates for integer matrices $A$ of exponents. Our main result is Theorem 6.10.

Given any $s=(s_{1},\ldots,s_{d})\in\mathbb{Z}^{d}$, denote by $\mathbb{R}^{d}_{s}$ the orthant

[TABLE]

In particular, $\mathbb{R}^{d}_{s}=\mathbb{R}_{>0}^{d}$ if $s\in 2\mathbb{Z}^{d}$. Let $x\in(\mathbb{R}^{*})^{d}$ be a solution of  (1.1). Then $x\in\mathbb{R}^{d}_{s}$ for some $s\in\mathbb{Z}^{d}$ (which is unique up to adding a vector in $2\mathbb{Z}^{d}$). Setting $z_{i}=(-1)^{s_{i}}x_{i}$, we get that $z=(z_{1},\ldots,z_{d})$ is a positive solution of the system with exponent matrix $A$ and coefficient matrix $C_{s}$ defined by $(C_{s})_{ij}=(-1)^{\langle s,a_{j}\rangle}c_{ij}$. Moreover, if $D$ is a Gale dual matrix of $C$, then the matrix $D_{s}$ defined by $(D_{s})_{ij}=(-1)^{\langle s,a_{i}\rangle}d_{ij}$ is a Gale dual matrix of $C_{s}$. Denote by $P_{i,s}$ the $i$-th row vector of $D_{s}$. Thus $P_{i,0}=P_{i}$ ($i$-th row of $D$) and $P_{i,s}=(-1)^{\langle s,a_{i}\rangle}P_{i}$, $i=1,\ldots,n$. Denote by ${{\mathcal{C}}}_{P_{s}}$ the positive cone generated by $P_{i,s}$ for $i=1,\ldots,n$.

Let ${\mathcal{M}}_{P}$ denote the complement in $\mathbb{R}^{k+1}$ of the hyperplane arrangement given by the hyperplanes $\{y\in\mathbb{R}^{k+1}\,:\,\langle P_{i},y\rangle=0\}$, $i=1,\ldots,n$. For any $\varepsilon\in\mathbb{Z}^{n}$ denote by ${\mathcal{C}}^{\nu}_{\varepsilon}$ the connected component of ${\mathcal{M}}_{P}$ defined by

[TABLE]

Note that ${\mathcal{C}}_{0}^{\nu}={\mathcal{C}}_{P}^{\nu}$.

Write $A^{\prime}$ for the matrix with column vectors $a_{1},\ldots,a_{n}$ ($A^{\prime}$ is obtained by removing the first row of $A$). It is convenient to introduce the map $\psi:\mathbb{Z}^{d}\rightarrow\mathbb{Z}^{1\times n}$ defined by

[TABLE]

where $s\in\mathbb{Z}^{d}$ is considered a row vector, i.e, an element of $\mathbb{Z}^{1\times d}$. Then, for any integer vector $b\in\ker(A)$ we have:

[TABLE]

Thus, applying Theorem 2.5 to the system with coefficient matrix $C_{s}$ and exponent matrix $A$, we obtain that the real solutions of  (1.1) contained in the orthant $\mathbb{R}^{d}_{s}$ are in bijection with the solutions of (2.8) in the quotient $\mathbb{P}{\mathcal{C}}^{\nu}_{\psi(s)}$ of the open cone $\mathcal{C}_{\psi(s)}^{\nu}$ by the equivalence relation $\sim$ defined in Section 2: $y\sim y^{\prime}$ if and only if there exists $\alpha>0$ such that $y=\alpha y^{\prime}$. In fact, such a bijection is given by the map which associates to any solution $x\in\mathbb{R}^{d}_{s}$ of the system (1.1) the unique $y\in\mathbb{R}^{k+1}$ such that $x^{a_{i}}=\langle P_{i},y\rangle$ for $i=1,\ldots,n$. We have proven the following result:

Proposition 6.5**.**

For any $s\in\mathbb{Z}^{d}$, there is a bijection between the real solutions of (1.1) contained in $\mathbb{R}_{s}^{d}$ and the solutions of (2.8) in $\mathbb{P}{\mathcal{C}}^{\nu}_{\psi(s)}$. This induces a bijection between the solutions of  (1.1) in $\mathbb{R}_{s}^{d}$ and the solutions of (2.8) in $\Delta_{P_{s}}={\mathcal{C}}^{\nu}_{\psi(s)}\cap\{y_{0}=1\}$ when $(1,0,\ldots,0)$ lies in the closure of the cone ${{\mathcal{C}}}_{P_{s}}$.

If $M$ is any matrix or vector with integer entries, we denote by $[M]_{2}$ the matrix or vector with coefficients in the field $\mathbb{Z}/2\mathbb{Z}$ obtained by taking the image of each entry by the quotient map $\mathbb{Z}\rightarrow\mathbb{Z}/2\mathbb{Z}$. Note that the following relation between the ranks holds: ${\rm rk}([A]_{2})={\rm rk}([A^{\prime}]_{2})$ if $[(1,1,\ldots,1)]_{2}$ belongs to the row span of $[A^{\prime}]_{2}$ and ${\rm rk}([A]_{2})={\rm rk}([A^{\prime}]_{2})+1$ otherwise. The following result is straightforward.

Lemma 6.6**.**

For any $s,s^{\prime}\in\mathbb{Z}^{d}$, we have ${\mathcal{C}}^{\nu}_{\psi(s)}={\mathcal{C}}^{\nu}_{\psi(s^{\prime})}$ if and only if $[s^{\prime}-s]_{2}$ belongs to the left kernel of $[A^{\prime}]_{2}$. For each $s\in\mathbb{Z}^{d}$, there are $2^{d-{\rm rk}([A^{\prime}]_{2})}$ distinct orthants $\mathbb{R}^{d}_{s^{\prime}}$ such that ${\mathcal{C}}^{\nu}_{\psi(s)}={\mathcal{C}}^{\nu}_{\psi(s^{\prime})}$.

Recall that since $A$ contains a row of ones, each polynomial in (2.8) is homogeneous of degree [math], which implies the following fact.

Lemma 6.7**.**

For any $\varepsilon\in\mathbb{Z}^{n}$, the map $y\mapsto-y$ induces a bijection between the solutions of (2.8) in ${\mathcal{C}}^{\nu}_{\varepsilon}$ and the solutions of (2.8) in ${\mathcal{C}}^{\nu}_{\varepsilon+(1,1,\ldots,1)}$.

Choose a $\mathbb{Z}$-Gale dual matrix $B\in\mathbb{Z}^{n\times k}$ of $A$ and consider the Gale dual system (2.8) it defines, for a given Gale dual matrix $D$ of a full rank matrix $C$. Since $B$ is a $\mathbb{Z}$-Gale dual matrix of $A$, we have $[A]_{2}\cdot[B]_{2}=0$ and furthermore the column vectors of $[B]_{2}$ are linearly independent over $\mathbb{Z}/2\mathbb{Z}$ because the greatest common divisor of the maximal minors of $B$ is equal to $1$. Therefore, $[B]_{2}$ is a Gale dual matrix of $[A]_{2}$ if ${\rm rk}([A]_{2})=d+1$. If the corank $c=d+1-{\rm rk}([A]_{2})$ is positive, then a Gale matrix of $[A]_{2}$ can be obtained by adding $c$ column vectors to $[B]_{2}$.

Define the feasible set $\Omega$ as follows:

[TABLE]

Equivalently, $\Omega$ is the set of all $\varepsilon\in\mathbb{Z}^{n}$ such that $[\varepsilon]_{2}$ belongs to the left kernel of any Gale dual matrix of $[A]_{2}$.

Proposition 6.8**.**

Let $\varepsilon\in\mathbb{Z}^{n}$. If $\varepsilon=\psi(s)$ for some $s\in\mathbb{Z}^{d}$, then $\varepsilon\in\Omega$. Conversely, assume that (2.8) has a solution in ${\mathcal{C}}_{\varepsilon}^{\nu}$. If ${\rm rk}([A]_{2})=d+1$ then $\varepsilon\in\Omega$. Assume that $\varepsilon\in\Omega$ if $c=d+1-{\rm rk}([A]_{2})>0$. Then, the following holds.

1. (1)

If ${\rm rk}([A]_{2})={\rm rk}([A^{\prime}]_{2})$, there exists $s\in\mathbb{Z}^{d}$ such that $[\varepsilon]_{2}=[\psi(s)]_{2}$. 2. (2)

If ${\rm rk}([A]_{2})>{\rm rk}([A^{\prime}]_{2})$, either there exists an integer vector $s\in\mathbb{Z}^{d}$ such that $[\varepsilon]_{2}=[\psi(s)]_{2}$, or there exists $s\in\mathbb{Z}^{d}$ such that $[\varepsilon]_{2}+[(1,1,\ldots,1)]_{2}=[\psi(s)]_{2}$. Moreover, there do not exist $s,s^{\prime}\in\mathbb{Z}^{d}$ such that $[\psi(s^{\prime})]_{2}=[(1,1,\ldots,1)]_{2}+[\psi(s)]_{2}$, so that only one of the two previous cases occurs.

Proof.

Let $y$ be a solution of (2.8) such that $y\in{\mathcal{C}}^{\nu}_{\varepsilon}$. Then, there exist positive real numbers $d_{i}$ such that $\langle P_{i},y\rangle=(-1)^{\varepsilon_{i}}d_{i}$ for $i=1,\ldots,n$ and using (2.8) this gives that $\sum_{i=1}^{n}\varepsilon_{i}b_{ij}$ is an even integer number for $j=1,\ldots,k$. Therefore, $[\varepsilon]_{2}$ belongs to the left kernel of $[B]_{2}$, which means that $\epsilon\in\Omega$ when ${\rm rk}([A]_{2})=d+1$. Assume that $\varepsilon\in\Omega$ if $c=d+1-{\rm rk}([A]_{2})>0$. Then, we get that $[\varepsilon]_{2}$ belongs to the left kernel of any Gale dual matrix $B_{2}$ for $[A]_{2}$. This left kernel is the image of the map $\mathbb{Z}^{d+1}\rightarrow{(\mathbb{Z}/2\mathbb{Z})}^{1\times n}$ sending $(s_{0},s_{1},\ldots,s_{d})$ to $[(s_{0},s_{1},\ldots,s_{d})]_{2}\cdot[A]_{2}=[s_{0}(1,1,\ldots,1)]_{2}+[\psi(s)]_{2}$, where $s=(s_{1},\ldots,s_{d})$. The image of this map is equal to the image of the map $s\mapsto[\psi(s)]_{2}$ precisely when ${\rm rk}([A]_{2})={\rm rk}([A^{\prime}]_{2})$. Thus item (1) and the first part of item (2) are proved. We also get that if $\epsilon=\psi(s)$ then $[\epsilon]_{2}$ belongs to the left kernel of $B_{2}$, and thus $\varepsilon\in\Omega$. Finally, note that if ${\rm rk}([A]_{2})>{\rm rk}([A^{\prime}]_{2})$ then there do not exist distinct $s,s^{\prime}\in\mathbb{Z}^{1\times(d)}$ such that $[\psi(s^{\prime})]_{2}=[(1,1,\ldots,1)+[\psi(s)]_{2}$ for otherwise $[(1,\ldots,1)]_{2}$ would belong to the row span of $[A^{\prime}]_{2}$. ∎

Example 6.9**.**

If $a_{1},\ldots,a_{n}\in 2\mathbb{Z}^{d}$, then ${\rm rk}([A^{\prime}]_{2})=0$ and ${\rm rk}([A]_{2})=1$. Moreover, the number of real solutions of  (1.1) is $2^{d}$ times its number of positive solutions. This number is equal to the number of solutions of (2.8) in the quotient cone $\mathbb{P}{\mathcal{C}}^{\nu}_{0}=\mathbb{P}{\mathcal{C}}_{P}^{\nu}$ by Theorem 2.5.

As a direct consequence of Proposition 6.5, Proposition 6.8, Lemma 6.6 and Lemma 6.7, we get the following result.

Theorem 6.10**.**

If ${\rm rk}([A]_{2})=d+1$, then there exists a solution of (1.1) in $(\mathbb{R}^{*})^{d}$ if and only if there exists a solution of (2.8) in ${\mathcal{M}}_{P}$. More generally, when ${\rm rk}[A]_{2}$ is not necessarily equal to $d+1$, there exists a solution of (1.1) in $(\mathbb{R}^{*})^{d}$ if and only if there exists a solution of (2.8) in a cone ${\mathcal{C}}_{\varepsilon}^{\nu}$ with $\varepsilon\in\Omega$.

Moreover, the number of solutions of (1.1) in $(\mathbb{R}^{*})^{d}$ is equal to $2^{d-{\rm rk}([A]_{2})}$ times the sum of the number of solutions of (2.8) in the quotient cones $\mathbb{P}{\mathcal{C}}^{\nu}_{\varepsilon}$, over all $\varepsilon\in\Omega$.

Recall that if $\varepsilon\in\Omega$, then $\varepsilon+(1,\ldots,1)\in\Omega$ because $(1,\dots,1)$ is assumed to be in the row span of $A$ (in fact, we are assuming that this is a row of $A$). Thus, the total number of solutions of (2.8) in the quotient cones $\mathbb{P}{\mathcal{C}}^{\nu}_{\varepsilon}$ over all $\varepsilon\in\Omega$ is even due to Lemma 6.7 (the solutions come in pairs of opposite real numbers).

Therefore, the number of solutions of (1.1) in $(\mathbb{R}^{*})^{d}$ is always even if ${\rm rk}([A]_{2})\leq d$ by Theorem 6.10. But note that when ${\rm rk}([A]_{2})=d+1$, then $2^{d-{\rm rk}([A]_{2})}=\frac{1}{2}$, and so in this case the number of solutions of (1.1) in $(\mathbb{R}^{*})^{d}$ is equal to half the sum of the number of solutions of (2.8) in the quotient cones $\mathbb{P}{\mathcal{C}}^{\nu}_{\varepsilon}$ over all $\varepsilon\in\mathbb{Z}^{n}$. In fact, by the second item in Proposition 6.8, when ${\rm rk}([A]_{2})=d+1$ there is a bijection between the solutions of (1.1) in $(\mathbb{R}^{*})^{d}$ and the solutions of (2.8) outside the hyperplanes $\{y\in\mathbb{R}^{k+1}\,:\,\langle P_{i},y\rangle=0\}$ in the real projective space $\mathbb{R}P^{k}$. The latter result is a consequence of [5, Theorem 2.1].

Given $A$ and $C$ and a choice of Gale dual matrices $B,D$, we saw in the proof of Theorem 3.8, that under the hypotheses of the theorem, it follows from Theorem 2.5 that $n_{\mathcal{A}}(C)>0$ is indeed equivalent to the existence of a solution to (3.1) in $\Delta_{P}$. In the previous sections, we have given different sufficient conditions on $D$ and $B$ such that system (3.1) has at least one solution in $\Delta_{P}$. When $A$ has integer entries it is then enough to check if these sufficient conditions are satisfied by $B$ and any matrix $D_{\varepsilon}$ obtained by multipliying the $i$-th row of $D$ by $(-1)^{\varepsilon_{i}}$ for some $\varepsilon\in\Omega$. In this case, (1.1) has at least one solution in $(\mathbb{R}^{*})^{d}$ by Theorem 6.10.

Acknowledgement

We are grateful to Alessandro A. Grande for raising a question that lead to the improvement of Section 3.