1 Introduction

Ehrhart -coefficients of polytopal complexes are non-negative integers

Abstract.

The Ehrhart polynomial of an integral polytope counts the number of integer points in integral dilates of . Ehrhart polynomials of polytopes are often described in terms of their Ehrhart -vector (aka Ehrhart -vector), which is the vector of coefficients of with respect to a certain binomial basis and which coincides with the -vector of a regular unimodular triangulation of (if one exists). One important result by Stanley about -vectors of polytopes is that their entries are always non-negative. However, recent combinatorial applications of Ehrhart theory give rise to polytopal complexes with -vectors that have negative entries.

In this article we introduce the Ehrhart -vector of polytopes or, more generally, of polytopal complexes . These are again coefficient vectors of with respect to a certain binomial basis of the space of polynomials and they have the property that the -vector of a unimodular simplicial complex coincides with its -vector. The main result of this article is a counting interpretation for the -coefficients which implies that -coefficients of integral polytopal complexes are always non-negative integers. This holds even if the polytopal complex does not have a unimodular triangulation and if its -vector does have negative entries. Our main technical tool is a new partition of the set of lattice points in a simplicial cone into discrete cones. Further results include a complete characterization of Ehrhart polynomials of integral partial polytopal complexes and a non-negativity theorem for the -vectors of rational polytopal complexes.

Key words and phrases:
Ehrhart theory, -vector, -vector, Ehrhart polynomial, counting interpretation, non-negativity, partial polytopal complex, simplicial complex, discrete cone
2010 Mathematics Subject Classification:
52B20, 52B70, 05A10, 05A15, 05E45, 11C08
The author was supported by the DFG (Deutsche Forschungsgemeinschaft) grant BR 4251/1-1.

1. Introduction

For any set the Ehrhart function counts the number of lattice points in the -th dilate of for . Ehrhart’s theorem states that if is a lattice polytope then is a polynomial in and, by induction, the same holds for polytopal complexes with integral vertices. [1, 10, 11]

Recently, a number of articles have appeared that realize various combinatorial counting polynomials as Ehrhart functions of suitable polytopal complexes and then apply results from Ehrhart theory to prove theorems about these combinatorial functions. [3, 4, 8, 15] In particular, it is possible to obtain bounds on the coefficients of these polynomials in this way. [6] For this purpose, the coefficients with respect to the monomial basis are not always easiest to work with. There are other bases of polynomial space that give rise to coefficient vectors such as the - and -vectors that are more amenable to analysis. These are defined as follows.

Let be a polynomial in of degree at most . Then there exist coefficients and for such that

(1)

The coefficients and depend both on and on , so we will sometimes write and to make this dependency explicit. The vectors and are called the - and -vectors of and their entries are the - and -coefficients of , respectively. Note that the -vector also goes by the name of Ehrhart -vector. [18] Whenever we refer to the - or -vector of an integral polytope or polytopal complex , we mean the - or -vector of its Ehrhart polynomial . For more details on these vectors and, most importantly, the motivation for defining them we refer the reader to Section 2.3.

One famous result about -vectors is Stanley’s theorem which asserts that the -coefficients of the Ehrhart polynomial of an integral polytope are always non-negative integers. [17] Behind this theorem lies a beautiful interpretation, due to Ehrhart, of the -coefficients of the Ehrhart polynomial of a simplex as counting lattice points at various heights in the fundamental parallelepiped of the cone over the homogenization of . [10, 11]

While -vectors of integral polytopes are always non-negative, -vectors of integral polytopal complexes may well have negative entries. Moreover, polytopal complexes with negative -coefficients appear in natural combinatorial applications. For example, coloring complexes of uniform hypergraphs can have negative -coefficients. Their -vector, however, is always non-negative. See Section 2.6 and [7] for details.

Thus, the question arises whether this is always true: Do polytopal complexes always have a non-negative integral -vector? The purpose of this article is to give a positive answer to this question.

Our main result is a counting interpretation of the -vector of a simplex , in the spirit of the classic counting interpretation of the -vector of a simplex. Given a relatively open lattice simplex , the -vector counts the number of so-called atomic lattice points at different heights in the fundamental simplex of the cone over the homogenization of . More precisely:

Theorem 1.

Let be an open lattice simplex, let and let . Then counts the number of atomic lattice points in the half-open fundamental simplex of at level .

The definitions of the fundamental simplex, atomic lattice points and their level are given in Section 3. An open lattice simplex is the relative interior of a lattice simplex.

From this counting interpretation we can immediately obtain a complete characterization of the -vectors of integral partial polytopal complexes. Here, an integral partial polytopal complex is any set that can be written as the disjoint union of relatively open polytopes with integral vertices.

Theorem 2.

A vector is the -vector of some integral partial polytopal complex if and only if it is integral and non-negative.

In particular, this gives us the desired non-negativity result for -vectors of polytopal complexes.

Theorem 3.

Every integral polytopal complex, and in particular every lattice polytope, has a non-negative integral -vector.

The crucial point here is that the -vector is non-negative and integral even if the complex does not have unimodular triangulation and even if its -vector has negative entries. Note that non-negativity of the -vector follows automatically if the complex has a unimodular triangulation or if the -vector is non-negative. This means that Theorem 3 gives a new result only if the complex in question is non-convex and does not have a non-negative -vector. But, as we already mentioned, there are non-convex polytopal complexes with negative -coefficients that do appear in practical applications.

The key technical ingredient that goes into the above counting interpretation is the following partition of the set of lattice points in a simplical cone into “discrete cones”.

Theorem 4.

Let be linearly independent integer vectors in for . Then

(2)

where the union ranges over all atomic lattice points in the half-open fundamental simplex of and this union is disjoint.

Here denote the level of and refers to all non-negative linear combinations of the whereas refers to all non-negative, integral linear combinations of the . Again, we refer to Sections 2 and 3 for details.

Theorem 4 is much more general then necessary for Theorems 1, 2 and 3 and is the main technical result of this article. In particular, Theorem 4 can be used to obtain a counting interpretation and a non-negativity theorem in the rational case.

Theorem 5.

Let be an open rational simplex, let and be a positive integer such that is integral. There exist polynomials such that for all integers and with the Ehrhart function of satisfies . Then for all and all the -coefficient counts the number of atomic lattice points in the half-open fundamental simplex of at level with .

This counting interpretation yields a non-negativity theorem for the -vector just as in the integral case. The -vector of a rational polytopal complex is the vector of all numbers for and , see Section 5 for details.

Theorem 6.

Any rational partial polytopal complex has a non-negative integral -vector.

Interestingly, there is another variant of Theorem 5 that expresses the Ehrhart function of a rational simplex in terms of restricted partition functions. For our purposes the restricted partition function is given by

see Section 5 for details. Then Theorem 4 allows us to write the Ehrhart function of a rational simplex in terms of restricted partition functions in the following way.

Theorem 7.

Let be an open lattice simplex with vertices and let be minimal positive integers such that is integral for all . Then

for all where and denotes the number of atomic lattice points at level in the fundamental simplex of with . Here, for all .


This paper is organized as follows. In Section 2 we give some preliminary definitions, sketch a classic proof of the non-negativity of -vectors for polytopes and give an example of a natural simplicial complex with a negative -vector. In Section 3 we present the partition of the set of lattice points in an open simplicial cone into discrete subcones, which is the main technical result of this article. In Section 4 we use this partiton result to give a counting interpretation of the -coefficients of a simplex, prove the non-negativity of the -vector and give a complete characterization of the Ehrhart polynomials of integral partial polytopal complexes. Up to this point we have mainly worked with integral polytopes, to make the ideas behind the construction more transparent. However, most of our results apply to the rational case as well. In Section 5 we introduce -vectors of rational polytopes, give a counting interpretation, prove the non-negativity of the -vectors of rational partial polytopal complexes and relate Ehrhart functions of rational simplices to restricted partition functions.

2. Preliminaries

Note: A comprehensive definition of all notions from polytope theory, Ehrhart theory or generating function theory that we make use of is out of scope of this article. For any undefined terms we refer the reader to [1, 16, 20].

2.1. Geometry

A polytope is the convex hull of finitely many points. A supporting hyperplane of a polytope is a hyperplane such that is contained in one of the two corresponding closed half-spaces. A face of is the intersection of a supporting hyperplane with . By convention is a face of itself as well. The dimension of is the dimension of its affine hull. The faces of dimension 0 are called vertices.

A polytope is integral if all its vertices are elements of the integer lattice , where is the dimension of the ambient space. Integral polytopes are also called lattice polytopes. Two polytopes are lattice equivalent if there is an affine isomorphism of the ambient space with that induces a bijection on the integer lattice .

The relative interior of a polytope is the interior of taken with respect to its affine hull. We also use the term open polytope to refer to the relative interior of a polytope. When we speak of the faces of an open polytope, we mean the faces of its closure. Every polytope is the disjoint union of the relative interiors of its faces.

A simplex is the convex hull of finitely many affinely independent points. A simplex of dimension has exactly vertices. The standard simplex of dimension is the convex hull of standard unit vectors. An integral simplex is unimodular if it is lattice equivalent to a standard simplex.

A polytopal complex is a finite set of polytopes with the following two properties: 1) If and is a face of , then . 2) If , then and is a face of both and . The elements of are also called faces of . The dimension of is the maximum dimension of any face of . The support of is the union of all polytopes in . A polytopal complex is integral if all of its faces are integral.

A simplicial complex is a polytopal complex whose faces are simplices. A triangulation of a set is a simplicial complex whose support is . A simplical complex is unimodular if all of its faces are unimodular. Note that not all integral polytopes, not even all integral simplices, have a unimodular triangulation.

The -vector of a -dimensional simplicial complex is the vector where is the number of -dimensional faces of . The -vector of is the vector defined in terms of the -vector via

for where . Note that the -vector has one more entry than the -vector but is fixed.

2.2. Ehrhart theory

As mentioned in the introduction, our point of departure is Ehrhart’s theorem, which states that for any integral polytope there exists a polynomial such that

for all .

It is straightforward to see that Ehrhart’s theorem carries over to polytopal complexes. However, many applications go one step further and work with “partial” polytopal complexes instead, where some faces are missing. In particular, inside-out polytopes are examples of half-open polytopal complexes that are widely used in combinatorial applications of Ehrhart theory. [3, 4, 6] Let us now make precise what we mean by “partial” in this context.

As defined in the previous section, a (relatively) open polytope is the relative interior of a polytope. The vertices, faces and facets of an open polytope are defined to be the vertices, faces and facets of its closure. Note that thus the vertices of an open polytope are not contained in the open polytope. An open polytope is called integral if all its vertices are integral.

For any polytopal complex , the support of is the disjoint union of the relative interiors of all faces of . This motivates the following definition: A partial polytopal complex is a disjoint union of open polytopes. The difference between a polytopal complex and a partial polytopal complex is therefore simply that some of the relatively open faces of the polytopal complex (that would need to be included because a polytopal complex has to be closed under passing to faces) have been removed.

Two important special cases of partial polytopal complexes are the following.

A “half-open” polytope is a set of the form where is a polytope and the are faces of . Every half-open polytope is the support of some partial polytopal complex. The half-open simplices that we are going to meet in the next section are examples of this.

A relative simplicial complex is a set of simplices of the form where is a simplicial complex and is a subcomplex of . Relative simplcial complexes can be written as partial polytopal complexes. They appear, for example, in Steingrímsson’s construction of the coloring complex. [19] Relative polytopal complexes can be defined similarly and again they can be realized as partial polytopal complexes. Inside-out polytopes are examples of relative polytopal complexes. [5]

2.3. - and -vectors

Let us denote by the -dimensional standard simplex with open facets, i.e.,

It turns out that for and in particular where is the relative interior of the standard -dimensional simplex. This has the following immediate consequences for a -dimensional integral polytopal complex .

  1. If has a unimodular triangulation , then can be written as a disjoint union of relatively open unimodular simplices of varying dimension . Thus

    where the coefficients count the number of -dimensional relatively open unimodular simplices appearing in the disjoint union. In this case the -vector of the simplicial complex coincides with the vector of coefficients , which explains the name.

  2. If has a unimodular triangulation that can be written as a disjoint union of unimodular half-open simplices , of fixed dimension , then

    where the coefficients count the number of -dimensional relatively open unimodular simplices appearing in the disjoint union. In particular, if is a shellable1 complex that is a topological ball then the -vector of coincides with the vector of coefficients , which explains the name. Note that if is not a topological ball, then is non-zero in general and the - and -vectors may differ.

If does not have a unimodular triangulation, we can still define the - and -vectors of . In fact, we can define - and -vectors for arbitrary polynomials by proceeding as sketched in the introduction.

For any integer ,

is a polynomial in of degree . Moreover, both

form bases of the vector space of polynomials in of degree at most . Thus, for any non-negative integer and any polynomial of degree at most we can define vectors and by

We call the -vector of and the numbers the -coefficients of . Similarly, we call the -vector of and the numbers the -coefficients of .

At this point, it important to call attention to the following subtlety: depends on the choice of , whereas does not. More precisely, the -vector has the following property. Let be any polynomial and let be any two integers. Then for all . This statement is false for . Despite this difference, we are going to suppress and in our notation for both and whenever it is clear from context which and are meant.

Now that we have defined the - and -vectors of polynomials, we can define - and -vectors of polytopes (and more generally polytopal complexes) via the Ehrhart function.

Let denote a polytopal complex. Then the - and -vectors of are, respectively, defined by

where .

If we do not specify explicitly, it is understood that , that is, and .

2.4. Generating function point of view

Classically, the -vector is defined in terms of generating functions.

Proposition 8 (c.f. Lemma 3.14 in [1]).

If is a polynomial of degree at most , then

A similar statement can be made about the -vector.

Proposition 9.

If is a polynomial of degree at most , then

Proof.

The coefficient of in the Laurent expansion of is precisely , the number of lattice points in the -th dilate of a -dimensional unimodular simplex. Thus

which yields the desired identity. ∎

Corollary 10.

The - and -vectors of a polynomial satisfy

2.5. Counting interpretation for the -vector

Given linearly independent integer vectors we define the cone over the by

Instead of allowing real coefficients , we can also restrict ourselves to integral coefficients. In this way, we obtain the discrete cone over the which is

The fundamental parallelepiped of the cone is

The crucial property of the fundamental parallelepiped is that it tiles the cone. That is, the cone can be written of as the disjoint union of integral translates of the parallelepiped, where the translation vectors are precisely the elements of the discrete cone. In terms of the Minkowski sum, this can be written simply as:

In particular

(3)

This can be phrased in terms of multivariate generating functions. Consider the ring of generating functions in the variables and write for any integer point . Then

(4)

since is the multivariate generating function of . Note that the numerator is a finite sum, so that if all are non-negative, the numerator is in fact a polynomial.

Now, let be integers and let denote the vertices of an integral simplex in . By embedding into at height , we pass to the vectors with and

which, expressed in terms of generating functions, reads

(5)

Combining identities (4) and (5), substituting 1 for and substituting for we obtain

where is the number of lattice points with .

This completes the proof of Ehrhart’s classic interpretation of the -vector.

Theorem 11 (Ehrhart [10, 11]).

Let be linearly independent and let for . Let denote the -vector of the -dimensional simplex . Then

By virtue of the fact that polytpoes are convex, the fact that every integral polytope can be triangulated and using a clever irrational shifting argument to get rid of lattice points on lower-dimensional faces [2], this theorem can be extended to general lattice polytopes.

Theorem 12 (Stanley [17]).

Let be a -dimensional integral polytope. Then the -vector of is non-negative.

Our goal is now to obtain a similar counting interpretation, and, in particular, a similar non-negativity result for the -vector of polytopal complexes. Before we come to this, we present examples of polytopal complexes where the -vector has negative entries.

2.6. -vectors with negative entries

Stanley’s theorem tells us that in order to find -vectors with negative entries we have to look outside the class of integral polytopes. We are going to consider integral polytopal complexes instead.

Coloring complexes of graphs are a class of simplicial complexes that have been studied by a number of authors in recent years, see, e.g., [7, 9, 12, 13, 14, 19]. All coloring complexes of graphs have a non-negative -vector. A natural generalization are coloring complexes of hypergraphs. For details about this notion, we refer the interested reader to [7].

A hypergraph is a finite set of vertices, together with a set of edges. An edge is a set of vertices of cardinality at least two. A proper coloring of is a labeling of the vertices of with the property that every edge contains at least two vertices that have a different color . Let be the set of all vectors in that are not equal to the all-one and all-zero vectors. We can now define the simplicial complex which is called the coloring complex of as follows. is a face of if and only if 1) , 2) for any two vertices we have or componentwise and 3) there exists an edge such that for all vertices and all we have . Notice that an element of appears as a vertex of if and only if , viewed as a coloring of the vertices of with exactly two colors and , is an improper coloring.

As an example, we consider the hypergraph on vertex set with edges , and . The associated coloring complex is 3-dimensional. It consists of three 3-dimensional spheres that share a single 0-dimensional subsphere . The spheres are simplicial complexes which can also be obtained by taking the boundary complex of the 5-dimensional cube triangulated by the braid arrangement and removing the all-zero and all-one vertices (and all incident faces). Then, the -vector of is

which has a negative entry.

Intuitively speaking, the reason for the negative entry is that the complex consists of spheres that have an intersection of codimension strictly greater than 1. Further examples of hypergraph coloring complexes with negative entries in their -vector can be constructed in this way.

3. Parititoning a simplicial cone into discrete cones

As we have seen, (3) gives a partition of the set of lattice points in into discrete cones. This partition is ideally suited for the analysis of the -vector. To get our hands onto the -vector, however, we need a different partition, given in Theorem 4, which we are going to develop in this section. Theorem 4 is the main technical result of this article, as the counting results in subsequent sections can be derived from Theorem 4 in a straightforward fashion.

In order to prove this partition result, we first need a couple of definitions. The basic idea is illustrated in Figure 1.

For every real number there exist an integer and a real number such that

Note that if is not an integer then and . But if , then and . So we call and the skew integral and skew fractional part of , respectively. If is a vector, we use and to genote the vector of skew integral and skew fractional parts of the components of , respectively.

Given linearly independent integer vectors , we define the fundamental simplex generated by these vectors by

The half-open fundamental simplex is

We say a point is at level if with and define to be the level of . We denote by the set of all lattice points in at level .

We now define sets with the property that . The definition is inductive:

We call the lattice pionts in atomic. If for some atomic , then we also call atomic. If furthermore then note that for all we have

where the denote the standard unit vectors.

Similar to our definition of , we write to denote the level of , i.e., is the unique integer such that .

We write to denote the degree of : If there exists an index such that , then is defined to be the smallest such index. If there is no such index, we let .

So is atomic if and only if is integer and there does not exist a such that is integer, and

These definitions are illustrated in Figure 1.

Figure 1. This figure illustrates Theorem 4. (Note that some lattice points in the cone are not shown here.) Suppose we have three linearly independent vectors . The simplex is the fundamental simplex of the cone generated by . The half-open fundamental simplex is the intersection of the fundamental simplex with the interior of the cone. It is partitioned into three levels as indicated. The first level consists of all points below the hyperplane spanned by , including the points on the hyperplane. The second level contains all points between the hyperplanes spanned by and , respectively, excluding the points on the former but including the points on the latter and similarly for level 3. The set of atomic lattice points in the cone is defined inductively. In the figure, the solid points in the interior of the cone show atomic lattice points, whereas the empty circles show lattice points that are not atomic. All lattice points at level 1 in the open cone are atomic, as the point labeled in the figure. Because is atomic, none of the points for are atomic. Suppose is a lattice point in level 2 that cannot be reached from any atomic lattice point in level 1 by adding . Then is atomic and all points that can be reached from by adding or are not atomic. Similarly, for any atomic lattice point in level 3, all points that be reached by adding the first three generators to are not atomic. There are no atomic lattice points above level 3. The statement of Theorem 4 is that by placing -dimensional discrete cones at all atomic lattice points in level in this way, we obtain a partition of the set of lattice points in the interior of .

Despite their inductive definition, it turns out that atomic coefficent vectors have a simple characterization.

Lemma 13.

Let be a lattice point in the interior of . This means in particular that . Then:

  1. If is not atomic, then .

  2. If , then .

  3. If , then .

  4. If , then there exists an atomic such that .

  5. If , then is not atomic.

  6. If , then is not atomic. In particular, there are only finitely many atomic lattice points.

So in particular we have the following characterization of atomicity:

is atomic if and only if for all indices .

Or equivalently:

is atomic if and only if .

Proof of Lemma 13..

We proceed in several steps.

Part (1): If is not atomic, then .

We have to show that there exists an index such that . If is not atomic, then there exists an atomic with such that , i.e., there exists a non-negative integral vector such that with for all . As , for some . Thus and as desired.

Part (2): If , then .

We have and for all . Thus for all .

Part (3): If , then .

If , then, by the pigeonhole principle, there is an such that whence .

Part (4): If , then there exists an atomic such that .

Let . Let be a sequence of coefficient vectors with constructed recursively as follows. We start with . Given , we distinguish two cases.

  1. If , then we define the next element in our sequence as . In this case, and Note that by part (3).

  2. If , then we stop and is the last element of our sequence. Note that , as implies that by part (2).

By part (1), we know that is atomic as . By construction, we know that

whence

where for all and so

as desired. Note that as but . Note also that is integral, so that is a lattice point.

Part (5): If , then is not atomic.

By part (4), it follows that for , which means that is not atomic.

Part (6): If , then is not atomic. In particular, there are only finitely many atomic lattice points.

If , then by part (3) and so is not atomic by part (5). Since every level contains only finitely many lattice points, it follows that the total number of atomic lattice points is finite. ∎

After these preparations, we can now show Theorem 4, the partition theorem at the heart of this article.

Proof of Theorem 4.

First, we note that without loss of generality, we can assume . Next, we observe that the right-hand side is contained in the left-hand side of (2) by construction. So we only have to show that the left-hand side is contained in the right-hand side and that the union is disjoint.

The union is disjoint.

Let where and are atomic, and . Without loss of generality, we assume that .

Note that because and are integer vectors, and, as both and are atomic, for all and for all