Periods of Ehrhart coefficients of rational polytopes

Let P ⊆ Rn be a polytope whose vertices have rational coordinates. By a seminal result of E. Ehrhart, the number of integer lattice points in the kth dilate of P (k a positive integer) is a quasi-polynomial function of k — that is, a “polynomial” in which the coefficients are themselves periodic functions of k. It is an open problem to determine which quasi-polynomials are the Ehrhart quasi-polynomials of rational polytopes. As partial progress on this problem, we construct families of polytopes in which the periods of the coefficient functions take on various prescribed values.


Introduction
Let P ⊆ R n (n 2) be a convex 1 rational polytope -that is, the convex hull of finitely many points in Q n .By a famous theorem of Ehrhart [8], the number of integer lattice points in positive integer dilates kP of P is given by a quasi-polynomial function of k.In particular, there exist coefficient functions c P,i : Z → Q with finite periods such that The function ehr P : Z → Z defined by ehr P (x) := n i=0 c P,i (x)x i is the Ehrhart quasipolynomial of P. (We refer the reader to [2,13,21] for introductions to Ehrhart theory.) The motivation for this paper is the problem of characterizing the Ehrhart quasipolynomials of rational polytopes.It is well known that, if P is an integral polytope (meaning that all vertices are in Z n ), then the coefficients c P,i are constants and ehr P is simply a polynomial.Already in this case, the question of which polynomials are Ehrhart polynomials is difficult.Beginning with the pioneering work of Stanley [19,20], Betke & McMullen [4], and Hibi [14], many inequalities have been shown to be satisfied by the coefficients of Ehrhart polynomials of integral polytopes.(For recent work in this area, see [1,6,7,10,11,17,23,22] and references therein.)Indeed, in the 2-dimensional case, a 1976 result of Scott [18] completely characterizes the Ehrhart polynomials of convex integral polygons.Nonetheless, a complete characterization is not yet known for the Ehrhart polynomials of convex integral polytopes in dimension 3 or higher.
Much less is known about the characterization of Ehrhart quasi-polynomials in the nonintegral case.For example, even in dimension 2, we do not know which polynomials are the Ehrhart polynomials of nonintegral convex polygons [15] 2 .
In this paper, we approach the problem of characterizing Ehrhart quasi-polynomials by focusing on the possible periods of the coefficient functions c P,i appearing in equation (1).Define the period sequence of P to be (p 0 , p 1 , . . ., p n ), where p i is the (minimum) period of c P,i .Our question is thus: What are the possible period sequences of rational polytopes?
If the interior of P is nonempty, then the leading coefficient function c P,n is a constant equal to the volume of P, and so p n = 1.More generally, McMullen [16] showed that each coefficient period p i is bounded by the corresponding i-index of P. The i-index is the least positive integer m i such that every i-dimensional face of the dilate m i P contains an integer lattice point in its affine span.We call (m 0 , . . ., m n ) the index sequence of P.
McMullen proved the following.[16,Theorem 6]).Let P be an n-dimensional rational polytope with period sequence (p 0 , . . ., p n ) and index sequence (m 0 , . . ., m n ).Then p i divides m i for 0 i n.In particular, p i m i .
We will refer to the inequalities p i m i in Theorem 1.1 as McMullen's bounds.It is easy to see that the indices m i of a rational polytope satisfy the divisibility relations Beck, Sam, and Woods [3] showed that McMullen's bounds are always tight in the i = n − 1 and i = n cases.It is also shown in [3] that, given any positive integers m n | m n−1 | • • • | m 0 , there exists a polytope with i-index m i for 0 i n.Moreover, all of McMullen's bounds are tight for this polytope.This construction establishes the following.The period sequences realized by Theorem 1.2 must satisfy the inequalities In the following sections, we extend the set of known period sequences by constructing rational polytopes in which the period sequences do not satisfy these inequalities.Alternatively, our constructions may be thought of as examples in which a particular one of McMullen's bounds is arbitrarily far from tight.Our first main result is the following, which is proved in Section 3.
In Theorem 1.3, we achieve convex polytopes.In other cases, we are unable to find convex constructions and must consider nonconvex rational polytopes.In general, we call a topological ball B in R n a rational polytope (not necessarily convex) if B is a union i∈I P i of a finite family {P i : i ∈ I} of convex rational polytopes, all with the same affine span, in which every nonempty intersection P i ∩ P j , i = j, is a common facet of P i and P j .
Our second main result, proved in Section 4, is the construction of nonconvex polytopes with period sequences of the form (1, . . ., 1, p, 1).

Building blocks
In this section, we fix notation and recall results that will be used in the constructions below.We also establish Theorems 1.3 and 1.4 in the dimension n = 2 case.
We are interested in the period sequences of quasi-polynomials.This period sequence is invariant under addition of polynomials.Thus, it will be convenient to consider quasipolynomials f (x) and g(x) to be equivalent when f (x) − g(x) is a polynomial.In this case, we write f (x) ≡ g(x).In particular, if f (x) ≡ g(x), then f (x) and g(x) have the same period sequence.The chief convenience of this notation is that, if The constructions in the following sections depend on certain 2-dimensional polygons studied in [15].Let p be a positive integer and set q := p 2 − p + 1. (Typically, p will be the desired period of a coefficient function in the Ehrhart quasi-polynomial of a rational polytope.)Let ⊆ R be the closed segment [− 1 p , 0].Then the Ehrhart quasi-polynomial of has the form ehr (x) = 1 p x + c ,0 (x), where c ,0 has (minimum) period p.Let P ⊆ R 2 be the convex pentagon with vertices u + , u − , v + , v − , w, where (We write e i for the ith standard basis vector.)A key fact, proved in [15], is that the Ehrhart quasi-polynomials of P and are "complements" of each other in the sense that the periodic parts of their coefficients cancel when the quasi-polynomials are added together 3 .That is, ehr P (x) ≡ − ehr (x). (2) As a warm-up for the following sections, we recall how P and were used in [15] to construct a polygon with period sequence (1, p, 1).Let R ⊆ R 2 be the rectangle [−q, q] × , and consider the convex heptagon H := Conv(R ∪ P ) ⊆ R 2 .Note that R and P are rational polygons whose intersection is the lattice segment with endpoints u ± .Hence we can use equivalence (2) to compute that ehr H (x) ≡ (2qx + 1) ehr (x) + ehr P (x) That is, H has period sequence (1, p, 1).This establishes the n = 2 cases of both Theorem 1.3 and Theorem 1.4.
3 Convex polytopes with period sequence (1, p, 1, . . ., 1) Let a positive integer p 1 be given.Recall that we set q := p 2 − p + 1.We now prove Theorem 1.3 by constructing a convex rational polytope H n ⊆ R n with period sequence (1, p, 1, . . ., 1).Since the n = 2 case was established in the previous section, we assume that n 3.
A useful fact about equivalence (2) is that it continues to hold when we take i-fold pyramids over both P and .More precisely, let Q ⊆ R d be a polytope, and let Q be the embedded copy of Q in R d+1 defined by Q := (x, 0) ∈ R d+1 : x ∈ Q .Fix a point a ∈ Z d+1 with final coordinate equal to 1. Then Conv(Q ∪ {a}) is a (1-fold) pyramid over Q.By induction, for i 2, define an i-fold pyramid over Q to be a pyramid over an (i − 1)-fold pyramid over Q. Proposition 3.1.Let P and be the pentagon and line segment defined in the previous section, and let ∆(P ) and ∆( ) be i-fold pyramids over P and , respectively.Then ehr ∆(P ) (x) ≡ − ehr ∆( ) (x). (3) Proof.The Ehrhart series Ehr Q (t) of a rational polytope Q is the generating function of ehr Q (x).That is, Ehr Q (t) is the formal power series

It is well known that Ehr
Given generating functions F (t) and G(t) of quasi-polynomials f (x) and g(x), respectively, we write F (x) ≡ G(x) if f (x) ≡ g(x).Hence, equivalence (2) implies that Ehr P (t) ≡ − Ehr (t), and so Equivalence (3) follows by comparing coefficients of the series.
One example of an i-fold pyramid that we will have occasion to use is the simplex S n ⊆ R n−1 given by S n := Conv 0, − 1 p e 1 , e 2 , . . ., e n−1 , which is an (n − 2)-fold pyramid over .It is known that the period sequence of S n is (p, 1, . . ., 1).Indeed, up to a lattice-preserving transformation, S n is among the polytopes constructed by Beck et al. [3] to prove Theorem 1.2.
We also construct an (n − 2)-fold pyramid over the pentagon P as follows.Write P ⊆ R n for the embedded copy of P defined by P := {(x, 0, . . ., 0) ∈ R n : x ∈ P }.We set P n to be the pyramid Note that, by Proposition 3.1, ehr Sn (x) ≡ − ehr Pn (x). (4) Let W n ⊆ R n be the translated prism over the simplex S n defined by where [−q, q] ⊆ R is a closed segment.(The reason for the translation by −qe 2 is that it will make the convex hull below easy to analyze.)From the construction of W n , it follows that ehr Wn (x) = (2qx + 1) ehr Sn (x).
We can now construct a convex polytope H n ⊆ R n which, we will show, has the period sequence (1, p, 1, . . ., 1).Let (See Figure 1 for the case where n = 3 and p = 2 case.)That H n has the desired period sequence is a direct consequence of the following lemma.
Figure 1: The polytope H 3 in the case where p = 2.
Lemma 3.2.The polytope H n defined above is a union of the form where W n ∩ M n , M n , and M n ∩ P n are lattice polytopes.
Proof.Note that W n , respectively P n , has a facet perpendicular to the e 2 -axis.Write F W , respectively F P , for this facet.That is, Let M n := Conv(F W ∪ F P ).To prove that H n = W n ∪ M n ∪ P n , it suffices to prove the following two statements: 1.For each facet F = F W of W n , P n lies on the same side of the hyperplane supporting F as W n does.
2. For each facet F = F P of P n , W n lies on the same side of the hyperplane supporting F as P n does.
In other words, excepting F W and F P , no facet of W n is visible from a vertex of P n and vice versa [9, Section 22.3.1].
To prove statement (1), recall that W n is a prism over a simplex.From this, the required facet-defining inequalities are easily determined and shown to be satisfied by the vertices of P n .To prove statement (2), note that P n is a pyramid over P n−1 .Hence, every facet of P n is either the "base" copy of P n−1 or a pyramid over a facet of P n−1 with apex e n .Thus, the required facet-defining inequalities are again easily determined by induction and shown to be satisfied by the vertices of W n .
It is now straightforward to complete the proof of Theorem 1.3.In particular, we prove the following: Theorem 3.3.Let a positive integer p be given.Then the convex rational polytope H n ⊆ R n constructed above has period sequence (1, p, 1, . . ., 1).
The reason for the constraint on the dimension n in Theorem 1.4 is that our construction depends upon the existence of a solution to a particular system of Diophantine equations in n − 1 variables, namely, the so-called ideal Prouhet-Tarry-Escott (PTE) problem.More precisely, we require integers s 1 , . . ., s n−1 > 0 and t 1 , . . ., t n−2 > t n−1 = 0 such that where p k (x) is the power-sum symmetric function of degree k in n − 1 variables.Such solutions to system (5) are known to exist when the number of variables is between 2 and 10 (inclusive) or is 12 [5, Chapter 11]4 .No solution in 11 variables is known.Wright [24] conjectures that solutions exist for every number of variables 2. However, Borwein [5, p. 87] gives a heuristic argument suggesting that this would be surprising.
Proof.Let = [− 1 p , 0] ⊆ R, and let P ⊆ R 2 be the pentagon defined in Section 2. Let B n ⊆ R n be the rational polytope defined as follows: Hence, B n is a union of two rational polytopes whose intersection is an integer polytope.Thus, ehr Bn (x) ≡ (t j x + 1) ehr (x).
We now exploit the fact that the s i and t j solve system (5).Newton's identities relating the power-sum symmetric functions to the elementary symmetric functions imply that the s i and t j also solve the system Therefore, ehr Bn (x) ≡ s 1 • • • s n−1 x n−1 ehr (x) ≡ s 1 • • • s n−1 c ,0 (x)x n−1 .That is, all coefficient functions of ehr Bn (x) are constant except for the coefficient of x n−1 , which has period p, as desired.