Interlacing Ehrhart Polynomials of Reflexive Polytopes

It was observed by Bump et al. that Ehrhart polynomials in a special family exhibit properties similar to the Riemann {\zeta} function. The construction was generalized by Matsui et al. to a larger family of reflexive polytopes coming from graphs. We prove several conjectures confirming when such polynomials have zeros on a certain line in the complex plane. Our main new method is to prove a stronger property called interlacing.


Introduction
The aim of the article is to investigate relations among three classical mathematical objects: graphs, polytopes and polynomials. There are known constructions that Akihiro Higashitani is partially supported by JSPS Grant-in-Aid for Young Scientists (B) 26800015. Mateusz Michałek is supported by the Polish National Science Center Grant 2013/08/A/ST1/00804. associate to a graph a lattice polytope-the symmetric edge polytope [24,27,28]. Furthermore, to each lattice polytope P one associates the Ehrhart polynomial H P [8,11,15,31] that computes the number of lattice points in dilations of P. This is also the Hilbert polynomial of the normal toric variety Spec C[C], where C is the cone over P [14,32]. Both constructions are briefly recalled in Sect. 2.1.
Furthermore, the roots of Ehrhart polynomials are also an object of intensive studies [6,9,12,20]. There are many graphs for which these roots have a remarkable property: They lie on a line R = {z : Re(z) = − 1 2 }. Still, proving that this property holds for a family of graphs is often very hard. One of the first positive results was the case of the complete (1, n)-bipartite graphs (trees) proved independently by Kirschenhofer et al. [23,Thm. 3.4] and by Bump et al. [5,Thm. 4,Thm. 6]. In the former, the authors studied this family of polynomials in relation to finiteness results on the number of solutions of diophantine equations and in relation to Meixner polynomials. In the latter this family of polynomials was studied in the context of the local Riemann hypothesis. Indeed, while the polynomials f that we associate to trees are symmetric with respect to R, the polynomials f (−x) appear as Mellin transforms of Laguerre functions and have properties similar to the Riemann ζ function [5,Ch. 3]. Related results were obtained by Rodriguez-Villegas, who further speculated about relations to the Riemann hypothesis [30,Ch. 4]. In fact, the study of functions with similar properties to ζ (but simpler) and their zero loci goes back to Pólya [29] and some of his methods apply to particular cases of polynomials we study [5, p. 4,5].
While our methods work easily for trees, they extend to many other classes. Our new approach is based on the observation that the polynomials we obtain come in families with roots having stronger properties than just belonging to the line R. Here we refer to the theory of interlacing polynomials [17], briefly discussed in Sect. 2.2. This theory has attracted much attention recently since Marcus, Spielman and Srivastava used it to solve the Kadison-Singer problem as well as to show the existence of bipartite Ramanujan graphs of all degrees [25,26]. It turns out it is also very useful while studying roots of special Ehrhart polynomials. One of the examples is the following: Consider two polynomials associated to complete bipartite graphs (1, n) and (1, n + 1). If we go along R, the roots of these two polynomials interchange, i.e. we never encounter consecutively two roots of one polynomial. The story does not end with graphs. Indeed, other reflexive polytopes exhibit similar properties. In particular, among other families, we present proofs for classical root polytopes of type C [1] and the duals of Stasheff polytopes [18].
Our general method consists of the following steps: (i) Determine the Ehrhart polynomials for a family of graphs. Here we either rely on known results, or we apply a Gröbner degeneration of the associated toric algebra to a monomial ideal, reducing to a combinatorial problem-cf. Proposition 4.4. (ii) Find recursive formulas for the Hilbert series/Ehrhart polynomials. The recursive formulas may involve auxiliary polynomials. (iii) Deduce the interlacing property from the recursive formulas.
Our main results show that: • The polynomials H K n associated to complete graphs have all roots on R-Corollary 3.8. In particular, [24,Conj. 4.8] holds.
• H G has all roots on R for any bipartite graph of type (2, n)-Theorem 4.12. In particular, [24,Conj. 4.7] holds. • H G has all roots on R for any complete bipartite graph of type (3, n)-Theorem 4.9. • The polynomials H P associated to classical root polytopes of type C and to dual Stasheff polytopes have all roots on R.
Note that finding classes of polytopes whose Ehrhart polynomials have all roots on R was also suggested as an open problem in [11, 2.43]. Apart from already mentioned methods like interlacing polynomials, recursive relations and Gröbner degenerations, our research also relates to topics like orthogonal polynomial systems, hypergeometric functions and reflexive polytopes. We strongly believe that our techniques will be further applied to many other families of polynomials.

Graphs, polytopes and polynomials
Let us fix a graph G = (V, E) and a lattice Z |V | with basis elements e v for v ∈ V . The associated lattice polytope P G ⊂ R |V | is the convex hull of We note that each edge of G corresponds to two vertices of P G , and apart from the vertices P G contains one more lattice point 0 ∈ Z |V | . The construction above is most common when G is an oriented graph. However for nonoriented graphs (or put differently, for oriented graphs with the property that we obtain a much more symmetric situation. Recall that a lattice polytope P is reflexive if 0 is its only interior point and the dual polytope P ∨ is also integral. Furthermore, it is terminal if every lattice point on the boundary is a vertex. For (nonoriented) graphs G the polytope P G is reflexive and terminal [24,Prop. 4.2]. Reflexive polytopes are very important and appear not only in combinatorics and algebra but also play a prominent role in algebraic geometry, e.g., through mirror symmetry [3]. In particular, the associated toric variety is Gorenstein.
Given a lattice polytope P ⊂ R d we denote by H P its Ehrhart polynomial, i.e., If P = P G , to simplify notation we write H G for the associated Ehrhart polynomial. Furthermore, if G is a complete k-partite graph of type (a 1 , . . . , a k ) ∈ Z k + , we write H (a 1 ,...,a k ) .
If P is d-dimensional, then H P is of degree d and the Ehrhart series is a rational function: We use similar subscript notation for the Ehrhart series. The sequence (δ 0 , δ 1 , . . . , δ d ) is called the δ-vector of P. Given a δ-vector for P, the Ehrhart polynomial can be reconstructed as We are now ready to see how the duality for reflexive polytopes reflects the symmetry properties on the level of algebra. Proposition 2.1 (cf. [3,19]) Let P be a lattice polytope of dimension d, H P (m) = a d m d + a d−1 m d−1 + · · · + 1 its Ehrhart polynomial and δ(P) = (δ 0 , δ 1 , . . . , δ d ) its δ-vector. Then the following four conditions are equivalent: (a) P is a reflexive polytope; Property (c) of the previous proposition shows exactly (skew)symmetry around − 1 2 . In the following we will denote R a = {z ∈ C : Re(z) = a}. Our goal will be to show that polynomials we consider are not only symmetric, but have all of their roots on R − 1 2 . Thus, we will denote R = R − 1 2 .

Interlacing polynomials
The theory of interlacing polynomials has proved to be very useful in various areas of mathematics. It was a crucial ingredient for the construction of bipartite Ramanujan graphs of all degrees [25], the solution of the Kadison-Singer problem [26] as well as the proof of the Johnson conjectures [4] to name only a few. More related to our work, recently it was also used for proving the real rootedness of polynomials appearing as the numerator of certain Hilbert series [21]. For a comprehensive treatment of interlacing polynomials and their properties we refer to [17]. Definition 2.2 (interlacing) Let L = α + R · β be a line in C with α, β ∈ C and let f, g ∈ C[x] be univariate polynomials with d = deg f = deg g + 1. Assume that both f and g have all their zeros on L. Let α+t 1 ·β, . . . , α+t d ·β and α+s 1 ·β, . . . , α+s d−1 ·β with t i , s j ∈ R be the zeros of f and g respectively. We say that f is L-interlaced by g if (after possibly relabeling) we have

Lemma 2.3
Let L ⊆ C be a line, let f, g 1 , . . . , g r ∈ C[x] be monic polynomials and let λ 1 , . . . , λ r ∈ R be nonnegative real numbers. If f is L-interlaced by each g i , then f is L-interlaced by λ 1 g 1 + · · · + λ r g r .
Proof Apply an affine transformation that sends L to R and use [ for some a, b ∈ R, b < 0. Then the following are equivalent: Proof Look at the Sturm sequence associated to f and g, cf. [10, §2.2.2].
Assume that there is an identity for some d ∈ R and i = 1, 2, 3. Then the following are equivalent: Proof First note that we necessarily have a = −d. If we replace x by ix + d and divide by i deg f 1 , we get the first claim from the preceding lemma. In order to prove the second statement, we compute the next elements of the Sturm sequence: After possibly dividing by x we can assume that the polynomials f 1 (x + d), f 3 (x + d) and f 5 (x + d) are even. Replacing each occurence of x 2 by x we get Note that the g i have only real and nonpositive roots. Thus, by [17, Lem. 1.82] g 3 interlaces g 1 . This implies the claim.

Orthogonal polynomial sequences
The theory of orthogonal polynomials is a classical topic in mathematics. Definition 2.6 (positive-definite moment functional, orthogonal polynomial system) A linear function μ : Examples for OPS include the Hermite polynomials, the Laguerre polynomials and the Jacobi polynomials.

Then there exists a positive-definite moment functional μ with respect to which f j is an OPS if and only if c j is real and λ j > 0 for each j. Further, for any positive-definite moment functional μ there exists a unique monic OPS and it satisfies the recurrence relation above.
Note that in this case Lemma 2.4 implies that every f j has only real roots and is interlaced by f j−1 . This is also the content of the Separation Theorem, cf. [13, Thm. I.5.3]. This property was used in [5] to show that certain polynomials obtained from the Mellin transform of Hermite polynomials and polynomials H 1,n (−x) obtained from the Mellin transform of Laguerre polynomials have all their roots on the line R 1 2 .

Ehrhart polynomials of reflexive polytopes as orthogonal polynomial sequences
In this section we will be interested in sequences of reflexive polytopes (P j ) ∞ j=0 whose Ehrhart polynomials f j (x) = H P j (x) satisfy a recurrence relation for some a j , b j , c j ∈ Q. Note that by Proposition 2.1 and since the constant term of every Ehrhart polynomial is equal to one, we can write the relation as for some M j ∈ Q. If 0 ≤ M j ≤ 1, then it follows from Lemma 2.5 that every f j has all its roots on R and is R-interlaced by f j−1 and ( is in that case an OPS. Note that since it is well-known that 2x + 1 is the unique Ehrhart polynomial of a reflexive polytope of dimension one and that every two-dimensional reflexive polytope has the Ehrhart polynomial ax 2 + ax + 1, where a ∈ { i 2 : i = 3, 4, . . . , 9}, we can conclude that M 2 should be one of 3 8 , 4 8 , . . . , 9 8 . In the following examples we will see that the values 4 8 , 5 8 , 6 8 , 8 8 actually appear. Furthermore, as we will see the polytopes giving rise to these OPS come in interesting families appearing in different branches of mathematics.
In fact, direct computations show the following: Therefore, the Ehrhart polynomial of the cross polytope Cr d has all roots on R and Note that since the zeros of the polynomial d k=0 d k t k = (1 + t) k in t are all −1, we can also prove that H Cr d (m) has the roots on R by applying [30] as mentioned in the introduction. Moreover, the H Cr d (m) coincides with H G for any tree G with (d + 1) vertices.
This polytope is the dual polytope of the so-called Stasheff polytope (associahedron). For more detailed information, see, e.g., [18]. In [2], its Ehrhart polynomial is calculated as follows:

Therefore, H A d has all roots on R and H
Example 3.6 (Classical Root Polytope of Type C) Let C d be the convex hull of the root system of type C, i.e., {±e i : We conclude that H C d has all roots on R and H C d+1 is R-interlaced by H C d .
Remark 3.7 Let B d (resp. D d ) be the classical root polytope of type B (resp. D) of dimension d. In [7], B d and D d are also discussed. It is proved in [7, Thm. 1] that We see that the δ-vector of B d is not palindromic, so the roots of H B d are not distributed symmetrically with respect to R. The δ-vector of D d is palindromic, so the roots of H D d are distributed symmetrically with respect to R. However, they do not have to lie on R. Below we present how the roots of H B 6 (on the left) and H D 6 (on the right) are distributed.
As a consequence of Example 3.5, we also obtain the following: Proof The classical root polytope of type A, A d , is unimodularly equivalent to the symmetric edge polytope of complete graphs. Hence, Example 3.5 directly proves the assertion.

Remark 3.9
Orthogonal polynomial systems are also studied in relation to hypergeometric functions [22]. We note that the Ehrhart polynomials described in Examples 3.3, 3.4, 3.5 and 3.6 can be presented also in that language: where r F s denotes the hypergeometric function.

Ehrhart polynomials and bipartite graphs
As we have seen in the previous section, OPS provide strong methods to prove that zeros of polynomials lie on R. However, as conjectured, e.g., in [24,Conj. 4.7], there exist families with roots on R, but not giving rise to OPS. Indeed, one can check that the polynomials H 2,n do not satisfy the recurrence relations in Theorem 2.7. As we shall see, for this setting, the correct generalization of orthogonal polynomial systems are interlacing polynomials-cf. Lemmas 4.6, 4.7, 4.8, Theorem 4.9 and Conjecture 4.10.

Complete bipartite graphs of type (2, n) and (3, n)
In this section we show that the Ehrhart polynomials H 2,n and H 3,n of the symmetric edge polytope of the complete bipartite graph of type (2, n) and (3, n) respectively have all roots on the line R = R − 1 2 . First, we determine the corresponding Hilbert series. We start by degenerating the binomial ideal to the monomial one. For any complete bipartite graph K a,b let us order the vertices in each part v 1 < · · · < v a , w 1 < · · · < w b and further v i < w j . There is an induced order on edges: (v i , w j ) < (v i , w j ) if i < i or i = i and j < j , analogously for the other orientation and (v i , w j ) < (w i , v j ) for any i, j, i , j . Further we declare the unique interior point 0 of the associated polytope P a,b to be smaller than any point corresponding to the edges. In the following lemma we describe a Gröbner basis for the ideal of interest. For a general reference concerning Gröbner bases we refer to [32].

Lemma 4.1 The ideal I a,b with respect to the induced degrevlex order has a quadratic Gröbner basis corresponding to
The leading terms are presented on the left.
Proof We leave the easy proof as the exercise for the reader. A more general statement can be found in [27].
The previous lemma motivates the following definition. , f (a, b, k)) A directed bipartite graph without any subgraphs corresponding to leading terms in Lemma 4.1 is called correct . Let f (a, b, k) be the number of correct (a, b)-bipartite graphs with exactly k edges (counted with multiplicities).

Definition 4.2 (correct graph
By the standard degeneration argument to the initial ideal [16,Thm. 15.26] we obtain: The Ehrhart polynomial H a,b evaluated at k counts the number of  (a, b)-bipartite correct graphs with at most k edges (possibly repeated).
Despite this explicit combinatorial description, in general we do not know how to determine the Ehrhart polynomial. Still, in specific situations we may find combinatorial recursive relations and easily prove that a given formula satisfies them.

Proposition 4.4 The Hilbert series for the complete graph bipartite K 1,n equals
The Hilbert series for the complete bipartite graph K 2,n equals The Hilbert series for the complete bipartite graph K 3,n equals Proof The first part is well known, cf. Example 3.3. The second was stated in [24]. The proof of the third has two steps. In the first we determine a recursive relation the Ehrhart polynomial must satisfy. In the second we deduce the Hilbert series form it.
1) Our aim is to find a formula for f (3, n, k). We note that in every correct graph, any vertex apart from v 1 and w 1 is either a sink or a source vertex. To simplify the terminology (e.g., degree of the vertex) we consider the graphs we count as simple and directed edges as (positively) weighted.
There are f (3, n − 1, k) graphs for which v n is of degree 0. Consider a correct graph G. By removing v n and edges adjacent to it we obtain a (3, n − 1)-bipartite correct graphG. Let us introduce the notation to count different types of correct graphs according to whether w 3 and w 2 are outgoing or incoming.
We denote by f xy (3, n, j) where x, y ∈ {i, o, z} the number of correct (3, n) bipartite graphs with j edges such that w 3 is of type x (i.e. incoming, outgoing or of degree zero) and w 2 is of type y. From now on we count graphs for which v n is not of degree zero and as both cases are similar we assume it is outgoing. According to the type of G we sum up: n − 1, j). Here we count graphs for which w 3 is incoming inGnote that in this case the only possibility to obtain G is to add the edge (v n , w 3 ) (with multiplicity k − j), (3, n − 1, j). Here we multiply by (k − j + 1) as we can distribute the weight (k − j) among the edges (v n , w 3 ) and (v n , w 2 ). n − 1, j). Here we distribute among all three edges. (v) We obtain the same when v n is incoming, apart from the fact that we have to replace i and o in all formulas. Summing up all we get n − 1, j)).
2) To pass from the recursive relation to the Hilbert series we proceed as follows. First we note that the Ehrhart polynomial H 3,n (i) = i k=0 f (3, n, k) and equivalently f (3, n, j) = H 3,n ( j)−H 3,n ( j −1). Thus summing up the recursive relation we obtain: At this point we could substitute all the values on the right-hand side and conclude. This however involves a lot of nontrivial computation on binomial coefficients. A better way is to pass to the Hilbert series. Precisely we multiply both sides of the equality (4.1) by t i and sum up over all natural i obtaining: Here the four Hilbert series in (4.2) (in order) correspond exactly to four terms in the recursive relation (4.1). Now the claim of the proposition is reduced to verifying that: So far we have used recursive relations to determine Hilbert series. However, it is also useful to go the other way round and determine further recursive relations using the known formulas.
We end this section with the following conjecture which encapsulates many of our results.  H a 1 ,...,a k , H a 1 −1,a 2 ,...,a k R-interlace.
Our results confirm the conjecture for a 1 = · · · = a k = 1 and also k = 2, a 2 = 1, 2. Furthermore, numerical experiments suggest that H a 1 ,...,a k , H a 1 −1,a 2 ,...,a k R-interlace whenever a 1 ≥ 2. We have shown this to be true in the cases k = 2 and a 1 = 1, 2, 3 and checked for all graphs with at most 10 vertices. and do not R-interlace.

Bipartite graphs of type (2, n)
By a more careful study of the involved polynomials we will show in this section that in fact the Ehrhart polynomial of every (not just complete) bipartite graph of type (2, n) has all roots on R. This will be derived as a special case for a more general statement for a larger family of polynomials.

Theorem 4.12 Let G be a bipartite graph of type (2, n). Then all roots of H G belong to R.
For all natural numbers j < d there exists a polynomial H d of degree j all of whose roots α ∈ C satisfy Re(α) = − d− j 2 .
Proof The recursion above is equivalent to the following recursion on the H d j : where Applying the formula for the Hilbert series of H d j one can show that both sides of Equation (4.7) have the same Hilbert series.
It follows from Lemma 4.15 that F d j is an even or odd polynomial, depending on the parity of j. Thus, there are polynomials Proof We will prove (i). The proof of (ii) is verbatim the same. Since the real part of all the roots ofH d 2k is − d− j 2 , the zeros of F d 2k (x) are located on the imaginary axis. This implies that the roots of A d k must be real and nonpositive. We show the rest of the claim by induction on k. For k = 0 the statement is obviously true. Assume that A d k−1 and A d k are coprime and interlace. Then it is immediate from the identity Let d be a positive integer and c ∈ R. In the following we consider the polynomial Proof Using (4.6) one checks that

Lemma 4.18
Let d be a positive even integer and let a n be the coefficient of the linear term in F d n . For 1 ≤ n ≤ d − 3 odd we have a n+2 a n > −(n + 1) 2 + (d − 2)(n + 1) + d.
In particular, Proof The proof is by induction on n, with n = 1 following from a 3 /a 1 = 3d − 6 > 3d − 8. Let f n := a n+2 /a n . By Lemma 4.15 we have By induction we know that Hence,  Proof First note that This has a double zero at − 1 2 if and only if the linear term of  (a 1 , . . . , a d ) → (0, a 1 , a 1 + a 2 , . . . , a 1 + a 2 + · · · + a d ) ∈ {0} × Z d .
An integral point a belongs to kA * d if and only if any two coordinates of t (a) differ (in absolute value) by at most k, or put differently, the coordinates of t (a) belong to an interval [a, a+k] for some a ∈ Z. Notice that coordinates of t (a) belongs to an interval of length b if and only if they belong to k −b+1 intervals of type [a, a +k] for different a ∈ Z. Thus, summing as multisets all integral points in t (kA * d ) for k = 0, . . . , m we enumerate integral sequences (0, c 1 , c 2 , . . . , c d ), each one counted that many times as many intervals We believe that further nice results will be obtained in future for the associahedra and duals of graph polytopes.