Sparse Polynomial Systems with many Positive Solutions from Bipartite Simplicial Complexes

10/19/2015 ∙ by Frédéric Bihan, et al. ∙ IAE Savoie Mont Blanc Inria 0

Consider a regular triangulation of the convex-hull P of a set A of n points in R^d, and a real matrix C of size d × n. A version of Viro's method allows to construct from these data an unmixed polynomial system with support A and coefficient matrix C whose number of positive solutions is bounded from below by the number of d-simplices which are positively decorated by C. We show that all the d-simplices of a triangulation can be positively decorated if and only if the triangulation is balanced, which in turn is equivalent to the fact that its dual graph is bipartite. This allows us to identify, among classical families, monomial supports which admit maximally positive systems, i.e. systems all toric complex solutions of which are real and positive. These families give some evidence in favor of a conjecture due to Bihan. We also use this technique in order to construct fewnomial systems with many positive solutions. This is done by considering a simplicial complex with bipartite dual graph included in a regular triangulation of the cyclic polytope.

READ FULL TEXT VIEW PDF
POST COMMENT

Comments

There are no comments yet.

Authors

page 1

page 2

page 3

page 4

This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.

1 Introduction

Real solutions of multivariate polynomial systems are central objects in many areas of mathematics. Positive solutions (i.e. solutions all coordinates of which are real and positive) are of special interest as they contain meaningful information in several applications, e.g. robotics, optimization, algebraic statistics, etc. In the 70s, foundational results by Kouchnirenko [26], Khovanskii [27] and Bernshtein [2] have laid theoretical ground for the study of the algebraic structure of sparse polynomial equations in strong connection with the development of toric and tropical geometry. These breakthroughs opened the door to computational techniques for sparse elimination [43, 19, 13, 41, 14].

Let be a finite point configuration. We consider unmixed sparse polynomial systems with support . This means that

coincides with the set of exponent vectors

of the monomials appearing in each equation. Kouchnirenko’s theorem [26] states that the number of toric complex solutions (no coordinate is zero) which are non-degenerate (the Jacobian matrix of the system is invertible at the solution) is bounded by the normalized volume of the convex hull of .

Viro’s method [44] (see also [33, 42, 3] for instance) is one of the roots of tropical geometry and has been used with great success for constructing real algebraic varieties with interesting topological types. It allows to recover under certain conditions the topological type for close to of a real algebraic variety defined by a system whose coefficients depend polynomially on a positive parameter . Here we apply a version of Viro’s method which has already been used in [40]. Given any finite configuration , where , an unmixed real polynomial system with support can be written as , where is a real matrix of size called coefficient matrix. Given a regular triangulation of the convex-hull of associated with a height function , we look at the deformed system . For sufficiently small, the number of complex (resp., real, positive) toric solutions of this deformed system is at least the total number of complex (resp., real, positive) toric solutions of the sub-systems obtained by truncating the initial system to the -simplices of the triangulation. We note that as far as we are only concerned with positive solutions, this construction works in the same manner if we allow real exponent vectors i.e. if . If the triangulation is unimodular, which means that all -simplices have normalized volume one, then all these sub-systems are linear up to monomial changes of coordinates. Generically, a real linear system has one complex toric solution, which is in fact a real solution. Since the number of -simplices in any unimodular triangulation of the convex-hull of is equal to its normalized volume, this construction produces polynomial systems whose all toric complex solutions are real [40, Corollary 2.4].

One goal of the present paper was to analyze under which conditions this construction produces polynomial systems whose all toric complex solutions are positive. Such polynomial systems are called maximally positive in [3], where a conjecture about their supports has been proposed [3, Conjecture 0.6].

Main results. Consider a regular full-dimensional pure simplicial complex supported on a point configuration and a coefficient matrix of size which encodes a map . We call a -simplex of this simplicial complex positively decorated by if the kernel of a submatrix of corresponding to this simplex contains vectors all coordinates of which are positive. This condition can be read off from the signs of maximal minors of . Said otherwise, the simplices that are positively decorated can be identified on the oriented matroid associated to . Moreover, our construction produces a polynomial system whose number of positive solutions is at least the number of -simplices of a regular triangulation of the convex-hull of which are positively decorated by . Our first observation is that any simplicial complex supported on whose all -simplices are positively decorated has a dual graph which is bipartite. This leads us to investigate three types of full-dimensional pure simplicial complexes: balanced simplicial complexes have the property that their set of vertices is -coloriable (two adjacent vertices have different colors) [39, Section III.4]; positively decorated simplicial complexes are characterized by the existence of a coefficient matrix which positively decorates all their -simplices; finally, bipartite simplicial complexes are characterized by the property that their dual graphs are bipartite. We shall see that balancedness implies that the complex can be positively decorated, which in turn implies that the complex is bipartite. For triangulations, these three properties are equivalent. Consequently, if a point configuration admits a regular, unimodular and balanced triangulation, then our construction produces maximally positive polynomial systems with support .

In order to illustrate this result, we check that some classical families of polytopes, namely order polytopes, hypersimplices, cross polytopes and alcoved polytopes, admit regular unimodular balanced triangulations and thus provide point configurations supporting maximally positive systems. As a by-product, we verify that these classes of point configurations have a basis of affine relations with coefficients in : this gives evidence in favor of Bihan’s conjecture [3, Conjecture 0.6].

Interesting computational problems arise from this analysis: if is a full-dimensional pure simplicial complex in whose dual graph is connected, deciding if it is balanced or if its dual graph is bipartite is computationally easy. However, deciding if it is positively decorable seems to be a nontrivial problem, which can be restated as deciding the existence of a realizable oriented matroid verifying conditions given by the combinatorial structure of the complex. We also show that the problem of decorating a simplicial complex can be recasted as a low-rank matrix completion problem with positivity constraints.

We apply our results to the problem of constructing fewnomial systems with many positive solutions comparatively to their number of monomials. Let be the number of variables (and equations) and the total number of monomials of a polynomial system. As a particular case of more general bounds, Khovanskii [27] obtained an upper bound on the number of (non-degenerate) positive solutions of such a system which depends only on and . This bound was later improved by Bihan and Sottile [6] to some constant times . When , taking univariate polynomials with distinct variables provides a construction with many positive solutions, while for the record construction is due to Bihan, Rojas and Sottile [7]. We focus here in the case , where the best construction so far is to consider a system with quadratic univariate equations with distinct variables, yielding positive solutions. By considering a subsimplicial complex of a regular triangulation of the cyclic polytope, we construct a pure simplicial complex of dimension on vertices whose dual graph is bipartite. Its number of -simplices grows as . Consequently, if this simplicial complex is positively decorable, then there exists a system with at least that many positive solutions. For , we compute explicitly such decorations and we ask the question whether they exist for any . In particular, for this yields a system with monomials and positive solutions.

Related works. Configurations of points that support maximally positive systems (systems such that all toric complex solutions are positive) have been characterized when is the set of vertices of a simplex (see e.g. [4]) or when is a circuit, see [3]. When is any finite subset of , it follows from Descartes’ rule of signs that should coincide with the intersection of its convex-hull with . Based on these characterizations, Bihan conjectured that if is the support of a maximally positive polynomial system, then there is a basis of affine relations for with coefficients at most in absolute value [3]. By [29, Lemma 7.6], this is equivalent to saying that the homogeneous toric ideal associated to can be written , where is generated by binomials with exponents at most . Balanced regular triangulations have also been used in [35] in order to get lower bounds for the number of real solutions of Wronski polynomial systems. Namely, given a regular balanced triangulation of a convex polytope in , a Wronski polynomial system is an unmixed polynomial system where all polynomials have the form with and is a linear combination with positive coefficients of monomials with given color . Since the triangulation is bipartite, all its -simplices get a sign so that two adjacent -simplices have opposite signs. Soprunova and Sottile showed in [35]

that under certain conditions on the polytope the absolute value of the difference between the number of positive and negative odd normalized volume

-simplices of the triangulation provides a lower bound on the number of real solutions of any Wronski polynomial system associated to this triangulation. In fact, Sottile informed us that the existence of maximally positive Wronski polynomial systems when the support admits a regular balanced unimodular triangulation follows from [35, Lemma 3.9]. Notice that our construction can be used to produce maximally positive systems which are not Wronski polynomial systems. In general, triangulations need not be balanced, but under some conditions, they admit a minimal branched balanced covering. This has been investigated by Izmestiev and Joswig [21, 22] for combinatorial -manifolds. The connection between the oriented matroid defined by the matrix of coefficients and the number of positive solutions has been investigated by Müller et al. in [30, Theorem 1.5], where they give a sufficient condition on this oriented matroid for a sparse system to have at most one positive solution.

Organization of the paper. Section 2 focuses on simplices and describes the construction of Viro’s system and its relation with the oriented matroid associated to a given coefficient matrix . In Section 3, the relationship between balanced, positively decorated and bipartite simplicial complexes is investigated. Section 4 focuses on unimodular and regular triangulations of classical polytopes, and we identify classes of polytopes for which there exists such a triangulation which is balanced, yielding construction of maximally positive polynomial systems. In Section 5, we focus on the cyclic polytope and we propose a construction of a bipartite subsimplicial complex with many simplices. We finish in Section 6 by relating the problem of positive decorability of a simplicial complex with two computational problems: the existence of realizable oriented matroids satisfying a specific condition, and the low-rank matrix completion problem with extra positivity constraints.

Acknowledgements.

We are grateful to Alin Bostan and Louis Dumont for their help with the computation of diagonals of bivariate series and asymptotic estimates of the growth of their coefficients. We also thank Éric Schost, Frank Sottile and Bernd Sturmfels for helpful discussions.

2 Positively decorated simplices

We start by focusing on systems of equations in variables involving monomials. This case corresponds to simplices and they shall serve as building blocks which will be glued to form simplicial complexes. Such systems are equivalent to linear systems up to a monomial map and the positivity of their solution can be read off from the signs of the maximal minors of the matrix recording the coefficients of the system.

Definition 2.1.

A matrix with real entries is called oriented if all the values are nonzero and have the same sign, where is the determinant of the square matrix obtained by removing the -th column.

The terminology “oriented” follows from the fact that for , the signs of the minors determine orientations of the edges of a -dimensional simplex compatible with an orientation of the plane (see Figure 1). The following proposition is elementary, but it plays a central role in the sequel of the paper.

Proposition 2.2.

Let be a full rank matrix with real entries. Then the following statements are equivalent:

  1. the matrix is oriented;

  2. for any , is an oriented matrix;

  3. for any permutation matrix , is an oriented matrix;

  4. all the coordinates of any non-zero vector in the kernel of the matrix are non-zero and share the same sign;

  5. there exists such that the -th column vector of belongs to the interior of the negative cone generated by the other column vectors of ;

  6. for any , the -th column vector of belongs to the interior of the negative cone generated by the other column vectors of .

Proof.

The equivalence follows from Cramer’s rule. and are proved directly by instanciating and

to the identity matrix.

follows from

is a consequence of the fact that permuting the columns of is equivalent to permuting the coordinates of the kernel vectors. Finally, the equivalence between , and is obvious once we have noticed that a positive vector belongs to the kernel of if and only if assuming , where are the column vectors of . ∎

We let denote the set of variables . A solution of a system is called non-degenerate if all the functions are  at and the Jacobian matrix of is invertible at . Throughout the paper, denotes a finite point configuration, and the coordinates of these points are recorded in a matrix (we assume that an ordering of the points has been arbitrarily fixed). We let denote the matrix obtained by adding a first row whose entries are all . For , the shorthand stands for the monomial .

Proposition 2.3.

Let and

be a system of Laurent polynomials with real coefficients involving monomials. If is invertible, then the system has one non-degenerate solution in the positive orthant if and only if the matrix recording the coefficients of the system is oriented.

Proof.

To any invertible real matrix with columns , we associate the bijection of the positive orthant

Its inverse map is . Let be the linear system defined by

By Cramer’s rule, this system has a solution in the positive orthant if and only if the matrix is oriented. Since , the positive solutions of are in bijection with those of . ∎

Let be a finite point configuration, and assume that the convex-hull of is a full-dimensional polytope . Let be a regular triangulation of the convex hull of i.e. is a triangulation and is a convex function, linear on each simplex of , but not linear on the union of two different maximal simplices of . Regular triangulations are sometimes called coherent or convex in the literature. Let be a matrix with real entries. We say that positively decorates a -simplex if the submatrix of given by its columns is oriented.

Consider the following family of polynomial systems parametrized by a positive real number :

(2.1)

where

For each positive real value of , this system has support included in .

Theorem 2.4.

There exists such that for all the number of non-degenerate solutions of (2.1) contained in the positive orthant is bounded from below by the number of maximal simplices in which are positively decorated by .

Proof.

Let be the maximal simplices in which are positively decorated by . Let be the restriction of to for . The function is affine hence there exist and such that for any . Moreover, since is convex and not affine on the union of two distinct maximal simplices of , setting and we get

where and is a polynomial in whose coefficients are products of real numbers by positive powers of . Since is positively decorated by , by Proposition 2.3 the system has one non-degenerate positive solution. Let be a compact set in the positive orthant which contains all the non-degenerate positive solutions of the systems for . If is small enough, the sets , , are pairwise disjoint and each one contains at least one non-degenerate positive solution of the system (2.1). ∎

Recall that is the convex-hull of and that stands for the Euclidean volume of multiplied by . The triangulation is called unimodular if has integer entries and any maximal simplex verifies . Sturmfels [40] showed that if has integer entries and is unimodular then for small enough the system (2.1) has exactly non-degenerate solutions with non-zero real coordinates, and no other solution with non-zero complex coordinates. The following proposition shows that if moreover the triangulation is positively decorated, then all these solutions are positive. We stress that the existence of maximally positive systems for monomial supports admitting a regular, unimodular and regular triangulation was already proved in [35, Lemma 3.9] using Wronski systems.

Proposition 2.5.

If is a matrix with integer entries, is a unimodular regular triangulation and all its -simplices are positively decorated by , then for small enough the system (2.1) has exactly non-degenerate positive solutions, and no other solution with non-zero complex coordinates.

Proof.

By Theorem 2.4, the system (2.1) has at least non-degenerate solutions in the positive orthant for small enough. On the other hand, the system (2.1) has at most non-degenerate solutions with non-zero complex coordinates by Kouchnirenko Theorem [26]. ∎

Polynomial systems whose all non-degenerate solutions with non-zero complex coordinates are contained in the positive orthant are called maximally positive in [3]. We shall put a special focus on classical polytopes admitting maximally positive polynomial systems in Section 4.

3 Bipartite dual graphs and balanced triangulations

The aim of this section is to show how Theorem 2.4 may be used to construct systems of polynomials with prescribed support and many positive real solutions. Throughout this paper, by simplicial complex, we always mean a full-dimensionsal pure geometric simplicial complex embedded in , see [28, Definition 2.3.5].

Definition 3.1.

Let be a finite configuration of points, represented by a matrix . A positively decorated simplicial complex supported on is a pair , where is a simplicial complex whose vertex set is a subset of and is a matrix such that every submatrix of size corresponding to a -simplex in is an oriented matrix.

Throughout this paper, we represent a pure -dimensional simplicial complex as a finite set , where has cardinality . For a -simplex and a coefficient matrix associated to the point configuration , we let denote the submatrix of whose columns correspond to the vertices in . Let be a -simplex in with vertices , and let be the corresponding matrix. Let be any point in the interior of and let be the matrix with columns . Clearly, the oriented matroid defined by does not depend on the choice of . We say that two matrices of the same size define the same (resp., opposite) oriented matroid if they have the same bases and two bases corresponding to the same set of columns have determinant of the same (resp., opposite) sign.

Lemma 3.2.

The matrix is oriented. Therefore, a matrix positively decorates if and only if either and define the same oriented matroid, or and define opposite oriented matroids.

Proof.

Let be the matrix obtained by removing the -th column from the matrix with columns (in this order) . Clearly, . On the other hand, we compute that coincide for with the determinant of the matrix obtained by adding to a first row of ones. Thus, all have the same sign, which means that is oriented. It follows that positively decorates if and only if either for all , or for all . ∎

Lemma 3.3.

Let and be two -simplices in with a common facet. Assume that the simplicial complex is positively decorated by a matrix . If and define the same oriented matroid, then and define the opposite oriented matroids.

Proof.

Due to Proposition 2.2, we may permute simultaneously the columns of and so that corresponds to the columns and corresponds to the columns . Thus, the submatrix given by the columns of and the submatrix given by the columns of are oriented.

Choose two points and in the interior of and respectively which are symmetric with respect to the common facet with vertices . Then, and

have opposite signs (a hyperplane symmetry has negative determinant). This means that

and have opposite signs. On the other hand, and are equal. It follows that if and define the same oriented matroid, then and define the opposite oriented matroids.∎

The dual graph of a pure simplicial complex is the adjacency graph of its simplices of maximal dimension.

Proposition 3.4.

Let be a pure simplicial complex of dimension in . If there exists a matrix such that the pair is a positively decorated simplicial complex, then the dual graph of is bipartite.

Proof.

This is a straightforward consequence of Lemma 3.3. In fact, the proofs of Lemma 3.3 and Lemma 3.2 show how one can associate a sign to each -simplex of so that any two adjacent (with a common facet) such simplices have opposite signs. Let denote the matrix obtained by adding a first row of to , and let be the square submatrix corresponding to a -simplex of . Let be the submatrix of corresponding to . Since is positively decorated, all have the same sign . We define then as the sign of the product . If we denote by the matrix obtained by adding a first row of to , we immediately obtain that is the sign of . ∎

A large class of simplicial complexes with bipartite dual graphs is provided by balanced simplicial complexes.

Definition 3.5.

[39, Section III.4]A -coloration of a simplicial complex supported on is a map (or more generally a map from to a set with elements) such that for any edge of . A -dimensional simplicial complex supported on is called balanced if there exists such a coloration.

Such colorations are sometimes called foldings since they can extended to a map from to the -dimensional standard simplex which is linear on each -simplex of . Similarly, balanced triangulations are sometimes referred to as foldable triangulations, see e.g. [25] and references within.

Proposition 3.6.

Let be the -th canonical basis vector of and be the vector . Let be a simplicial complex supported on . If is balanced and is a -coloring of , then the matrix with columns positively decorates .

Proof.

By construction, every submatrix of corresponding to a -simplex of is a column permutation of the matrix with columns . This latter matrix is oriented and hence the statement follows from Proposition 2.2. ∎

A pure simplicial complex is called locally strongly connected if the dual graph of the star of any vertex is connected. By [23, Proposition 6] and [23, Corollary 11], a locally strongly connected and simply-connected complex on a finite set is balanced if and only if its dual graph is bipartite, see also [21, Theorem 5]. It is worth noting that any triangulation of (i.e. a triangulation of the convex-hull of with vertices in ) is locally strongly connected and simply connected.

Theorem 3.7.

Assume that a finite full-dimensional point configuration in admits a regular triangulation which is balanced, or equivalently, whose dual graph is bipartite. Then there exists a polynomial system with support whose number of positive solutions is at least the number of -simplices of . If furthermore and the triangulation is unimodular, then there exists a polynomial system with support which is maximally positive.

Proof.

This is a direct consequence of Theorem 2.4, Proposition 3.6 and Kouchnirenko’s theorem [26]. ∎

We finish this section by an explicit example of a polynomial system with prescribed number of positive solutions obtained using this construction.

Figure 1: The balanced simplicial complex from Example 3.8, a -coloration of its -skeleton, and the induced orientations of its edges.
Example 3.8.

Let , , . Choosing heights

provides us with a regular triangulation of which has the balanced subsimplicial complex described in Figure 1. Applying the construction of Theorem 2.4 and Proposition 3.6, we obtain the following system, depending on a parameter :

which has at least six solutions in the positive orthant for sufficiently small. Setting and using Gröbner bases library FGb [16] and the real solver rs_isolate_sys [34] in Maple confirms that this system has indeed six positive solutions.

At this point, we would like to recapitulate the relationship between the properties of simplicial complexes studied in this section. As discussed above, balanced simplicial complexes are always positively decorable (Proposition 3.6) and the dual graph of positively decorable simplicial complexes is necessarily bipartite (Proposition 3.4). In summary:

By results of Joswig [23], for locally strongly connected simplicial complexes which are simply connected, these three properties are equivalent. However, it is not the case for simplicial complexes which do not verify these assumptions as Figure 2 gives an example of a simplicial complex whose dual graph is bipartite but which is not balanced. The reader can verify that it is positively decorable, hence positive decorability is not equivalent to balancedness.

We do not know an example of a pure simplicial complex of dimension embedded in whose dual graph is bipartite but which is not positively decorable, and we therefore ask the following question.

Question 3.9.

Is a pure full-dimensional simplicial complex positively decorable if and only if its dual graph is bipartite? If not, construct a counterexample.

Figure 2: A two-dimensional simplicial complex whose dual graph is bipartite but which is not balanced. The white triangle is not a -simplex of the complex.

4 Polytopes and maximally positive systems

Next, we turn our attention to finite sets which are supports of maximally positive systems. We recall that maximally positive systems have all their toric complex solutions in the positive orthant.

4.1 Order polytopes

Here we recall the description given in [35], which is itself based on [37]. Let be any finite partially ordered set, a poset for short, on elements. A chain of is a subset which is totally ordered. Two subsets of are called incomparable if for all , the elements and are incomparable. The order polytope of is the set of points such that whenever in .

Example 4.1.
  1. If all elements of are incomparable, then .

  2. If is a totally ordered set with elements, then is a primitive -simplex. For instance, if with usual increasing order, then is the unit simplex with vertices the origin and for , where stands for the -th vector of the canonical basis.

  3. If with order given by and , then is the convex-hull of the set of points , , , , .

  4. If is a disjoint union of incomparable chains of lenghts , then the order polytope is isomorphic to a Cartesian product of unit simplices of dimensions .

  5. More generally, if is a disjoint union of incomparable sub-posets , then .

A linear extension of is an order-preserving bijection from to . To each linear extension of , we associate an unimodular -dimensional simplex defined by , where . The vertices of are and for , where stands for the canonical basis of . The simplices are the -dimensional simplices of a triangulation of called canonical triangulation. Two -dimensional simplices and in have a common facet if and only if there is transposition of such that . Fixing a linear extension of identifies each linear extension of with a permutation of , where the fixed linear extension is identified with the identity permutation. The sign of a simplex is then defined as the sign of the corresponding permutation. Note that this is defined up to the choice of a fixed linear extension, another choice eventually changes simultaneously all signs. Thus, the adjacency graph of -dimensional simplices of is bipartite. It follows then from [23, Corollary 11] that is balanced. In fact, this can be shown directly by noting that the map , (where is the number of non-zero coordinates of ) restricts to a -coloration (with values in ) on giving different values to a pair of adjacent vertices of . It turns out that the canonical triangulation is also regular. A convex function certifying the convexity of is given by [35, Lemma 4.6]. We now summarize these properties of .

Proposition 4.2.

[35] The canonical triangulation of is regular, unimodular, and balanced.

Theorem 4.3.

For any poset , there exists a real polynomial system with Newton polytope which is maximally positive.

Proof.

This follows readily from Theorem 3.7 and Proposition 4.2. ∎

We can be more explicit. As we already saw, the map is a -coloring map onto giving distinct values to adjacent vertices of . Then, as shown in the proof of Proposition 3.6, the triangulation is decorated by the matrix with column vector corresponding to and column vector for any other vertex of . We now use the convex function , which shows that is regular, to get the following Viro polynomial system.

(4.1)

By Proposition 2.5, for small enough the system (4.1) is maximally positive. For instance, if with partial order defined by , then is the convex-hull of the points , , , , , (a prism) and (4.1) may be written as .

Systems of multidegree are a special case of Theorem 4.3: their support is the order polytope of a disjoint union of incomparable chains (see Example 4.1, item 4). This implies the following statement.

Corollary 4.4.

Let be any positive integer and be any partition of into positive integers. Set , where for . Among multilinear (in other words multidegree ) polynomial systems in , there exists a maximally polynomial system.

There is another possible construction of maximally positive multilinear systems, which uses strictly totally positive matrices (i.e. matrices whose all minors of any size is positive). Note that such matrices exist [1, Thm 2.7]. The interest of the following proposition is that it gives a direct construction of maximally positive multilinear systems which does not depend on a parameter . However, it does not seem to generalize easily to multi-homogeneous systems, while the construction above does (we give a proof of this fact at the end of this section).

Proposition 4.5.

Let be any positive integer and be a partition of into positive integers. Then there exist strictly totally positive matrices of respective dimensions such that all toric complex solutions of the system lie in the positive orthant, where

Proof.

By Kouchnirenko’s theorem, the number of complex toric non-degenerate solutions is bounded by the multinomial coefficient . For any partition of the set into parts of respective sizes , we associate the linear system , where

By construction, divides , hence a solution of the linear system is a solution of . Next, note that the condition that the matrices are totally positive imply that the linear system has a unique solution in the positive orthant. There are possible partitions of into parts of respective sizes . Each of them yields one solution in the positive orthant. To finish the proof, we prove that there exist such that all these solutions are distinct. This is done by noticing that the set of -uples of strictly totally positive matrices is a non-empty open subset for the Euclidean topology of -uples of matrices with real entries, while the set of such tuples of matrices leading to coalescing solutions is Zariski closed and hence has Lebesgue measure . ∎

The vertices of

are characteristic functions of upper order ideals of

. The set of such upper order ideals ordered by inclusion is a distributive lattice. The toric ideal of the set of integer points of the order polytope is generated by binomials over all incomparable upper order ideals of , where and [18]. These binomials are homogeneous binomials with exponents at most  (in fact at most ). Thus Bihan’s conjecture (see the introduction) holds true for order polytopes.


We finish this section by reporting on other classical families of polytopes which admit unimodular balanced regular triangulations. We stress that in all the following cases, Bihan’s conjecture [3] holds.

Cross polytopes. The -dimensional cross polytope is the subset of defined by points verifying . It has normalized volume and has a regular unimodular triangulation obtained by slicing it along the coordinate hyperplanes for . Associating a sign to each of the simplex of the triangulation counting the parity of the number of negative coordinates provides a -coloring of its dual graph. By Proposition 3.6 this triangulation is balanced.

Products of balanced triangulations. If , are two point configurations which admit regular balanced unimodular triangulations, then the product of these triangulations yields a regular balanced subdivision of the convex hull of into products of simplices. Joswig and Witte show in [24] that this subdivision can be refined into a regular balanced unimodular triangulation. Consequently, if and admit regular balanced and unimodular triangulation, then so does .

Joins of balanced triangulations. If , are two full-dimensional polytopes admitting regular unimodular balanced triangulations, then the natural triangulation of the join by joins of full-dimensional simplices in the triangulations of and is regular and unimodular (see [17, Section 2.3.2]). The fact that this triangulation is balanced can be seen by coloring the vertices of the triangulation of by and the vertices of the triangulation of by .

Alcoved polytopes. Alcoved polytopes are polytopes whose codimension faces lie on hyperplanes of the form , with . Lam and Postnikov [31] showed that such polytopes admit a natural unimodular triangulation compatible with the subdivision of given by the complements of all the hyperplanes of the form , for . The fact that this triangulation of alcoved polytopes is regular and balanced is a special case of Lemma 4.6 below.

Let be a collection of vectors. This induces an infinite arrangement of hyperplanes for all and . Let be a -dimensional lattice polytope in . Assume that each -dimensional face of is contained in some hyperplane of . Consider the subdivision of obtained by slicing along the hyperplanes of .

Lemma 4.6.

The subdivision is regular and its dual graph is bipartite.

Proof.

The regularity of is obtained by considering the restriction to the vertices of of (see [17, Theorem 2.4]). It remains to show that the dual graph of is bipartite. In fact we show that the dual graph of the subdivision of obtained by slicing along the hyperplanes of is bipartite. For any given , associate a sign to each connected component of

so that adjacent components get opposite signs. Any connected component of is a common intersection of such connected components for , and we equip it with the product of the corresponding signs. Clearly, two adjacents components of have opposite signs. ∎

The hypersimplex. For , , the hypersimplex is the convex hull of all vectors in whose coordinates are ones and zeros. Several unimodular triangulations of the hypersimplex have been proposed. One is given by Sturmfels in [41]. Another one is described by Stanley in [38]. As explained in [31, Section 2.3], the hypersimplex is linearly equivalent to an alcoved polytope. Consequently, it admits a unimodular balanced and regular triangulation.

Multi-homogeneous systems. A multi-homogeneous system with respect to a partition and with multi-degrees is a system whose monomial support correspond to the set of lattice points in the polytope , where is the convex hull of in . Using the construction by Joswig and Witte [24], it is therefore sufficient to show that admits a regular balanced unimodular triangulation for any . Under the linear change of variables , we see that is defined by inequalities , for