Mixed Ehrhart polynomials

For lattice polytopes $P_1,\ldots, P_k \subseteq \mathbb{R}^d$, Bihan (2014) introduced the discrete mixed volume $\mathrm{DMV}(P_1,\dots,P_k)$ in analogy to the classical mixed volume. In this note we initiate the study of the associated mixed Ehrhart polynomial $\mathrm{ME}_{P_1,\dots,P_k}(n) = \mathrm{DMV}(nP_1,\dots,nP_k)$. We study properties of this polynomial and we give interpretations for some of its coefficients in terms of (discrete) mixed volumes. Bihan (2014) showed that the discrete mixed volume is always non-negative. Our investigations yield simpler proofs for certain special cases. We also introduce and study the associated mixed $h^*$-vector. We show that for large enough dilates $r P_1, \ldots, rP_k$ the corresponding mixed $h^*$-polynomial has only real roots and as a consequence the mixed $h^*$-vector becomes non-negative.


Introduction
Given a lattice polytope P ⊆ R d , the number of lattice points |P ∩ Z d | is the discrete volume of P . It is well-known ( [11]; see also [1,2]) that the lattice point enumerator E P (n) = |nP ∩Z d | agrees with a polynomial, the so-called Ehrhart polynomial E P (n) ∈ Q[n] of P , for all non-negative integers n. Recently, Bihan [5] introduced the notion of the discrete mixed volume of k lattice polytopes P 1 , . . . , P k ⊆ R d , where P J := j∈J P j for ∅ = J ⊆ [k] and P ∅ = {0}.
To the best of our knowledge, the origin of the mixed Ehrhart polynomial (2) goes back to Steffens and Theobald [22], see also [21]. In their work, a slight variant of (2), yet under the same name, has been employed as a very specific means to study higher-dimensional, mixed versions of Pick's formula in connection with the combinatorics of intersections of tropical hypersurfaces. The definition of the mixed Ehrhart polynomial therein differs from our definition by the exclusion of the empty set from the sum.
Here, we study mixed Ehrhart polynomials and their coefficients with respect to various bases of the vector space of polynomials of degree at most d. First, we show that in the usual monomial basis, the coefficients of mixed Ehrhart polynomials can be read off directly from the multivariate Ehrhart polynomial E P (n 1 , . . . , n k ), where n 1 , . . . , n k ∈ Z ≥0 are non-negative integers (Theorem 2.4). This gives a meaning to the coefficients of ME P 1 ,...,P k (n). In particular this provides a simple proof that the coefficient of n i vanishes for i < k and also allows to give streamlined proofs for known characterizations of the two leading coefficients (Corollaries 2.5 and 2.6). We then deal with two prominent subclasses.
For the case that all polytopes P 1 , . . . , P k are equal, the mixed Ehrhart polynomial can be expressed in terms of the h * -vector of P (see Proposition 2.7). And for the case that P 1 , . . . , P k all contain the origin and satisfy dim(P 1 + · · · + P k ) = dim P 1 + · · · + dim P k , we can provide a combinatorial interpretation of DMV(P 1 , . . . , P k ) (see Proposition 2.9). As a consequence of the main result from [5], it follows that ME P 1 ,...,P k (n) ≥ 0 for all n ≥ 0 (see Theorem 3.1). This is accomplished by the use of irrational mixed decompositions and skilled estimates. Using our understanding of the coefficients of ME P 1 ,...,P k (n), we give direct proofs for this fact in the cases k ∈ {2, d − 1, d} and P 1 = · · · = P k = P in Section 3. See also [14] for further developments in the context of valuations. Expressing the (usual or mixed) Ehrhart polynomial in the basis { n+d−i d : 0 ≤ i ≤ d} gives rise to the definition of the (usual or mixed) h * -vector and h * -polynomial (see Section 4). By a famous result of Stanley [18], the usual h * -vector is non-negative (componentwise). We illustrate that for the mixed h * -vector this is not true in general (see Example 4.3). Yet, we show that this has to hold asymptotically for dilates rP 1 , rP 2 , . . . , rP k (Corollary 4.6). This follows from the stronger result that for r 0 the mixed h *polynomial is real-rooted with roots converging to the roots of the d th Eulerian polynomial (Theorem 4.5). This can be seen as the mixed analogue of Theorem 5.1 in [10] (see also [3,7]). As a byproduct, we obtain that asymptotically the mixed h * -vector becomes log-concave, unimodal and, as mentioned, in particular positive, except for its 0 th entry, which always equals 0 (see Corollary 4.6). Our paper is structured as follows. In Section 2, we prove various structural properties of the mixed Ehrhart polynomial. In Section 3, we review Bihan's non-negativity result of the discrete mixed volume, particularly from the viewpoint of the mixed Ehrhart polynomial, and provide alternative proofs for some special cases. Finally, in Section 4 we study the h * -vector and the h * -polynomial of the mixed Ehrhart polynomial, and in particular show real-rootedness of the mixed h * -polynomial and positivity of the h * -vector for large dilates of lattice polytopes.

Structure of the mixed Ehrhart polynomial
In this section, we collect basic properties of the mixed Ehrhart polynomial. For some known results we provide new or simplified proofs. For the whole section, we fix a collection P = (P 1 , . . . , P k ) of k lattice polytopes in R d and we assume that P 1 + · · · + P k is of full dimension d. The mixed Ehrhart polynomial, as introduced in (2), is by definition a univariate polynomial of degree ≤ d, which can be written as ME P (n) = me d (P)n d + me d−1 (P)n d−1 + · · · + me 0 (P).
Example 2.1. For the case of k copies of the d-dimensional unit cube, P 1 = · · · = P k = [0, 1] d , we have E P i (n) = (n + 1) d and thus by the binomial theorem. Hence, where ∆ k f (0) denotes the k th difference of the function f at 0 (see, e.g., [19,Sect. 1.4]). This can be expressed as me i (P) = d i k!S(i, k) in the Stirling numbers S(i, k) of the second kind (see [19,Prop. 1.4.2]). Remark 2.2. If P = (P 1 , . . . , P k ) is a collection of lattice polytopes in R d and dim P i = 0 for some i, then ME P (n) = 0. To see this, assume that P k is 0-dimensional. For any J ⊆ [k−1], the polytopes P J and P J∪{k} are translates of each other and the corresponding terms in (2) occur with different signs.
As our first result, we give a description of the coefficients of the mixed Ehrhart polynomial in Theorem 2.4. For this we recall a result independently due to Bernstein and McMullen; see [13,Theorem 19.4].
The proof also recovers Lemma 4.10 from [22] with different techniques. In order to give an interpretation to some of the coefficients of ME P (n), let us write where E i P (n 1 , . . . , n k ) is homogeneous in n 1 , . . . , n k of degree i. By the same reasoning as in the case of the usual Ehrhart polynomial (see, e.g., [2, Sect. 3.6]), we note In the above expression, the (normalized) coefficient MV is called the mixed volume; see, e.g., [13,Chapter 6]. Moreover, the notation MV d (P 1 [α 1 ], . . . , P k [α k ]) means that the polytope P i is taken α i times, and (4) comes from a well-known property of mixed volumes (see [17,Section 5.1]). Based on (4) together with Theorem 2.4, we obtain streamlined proofs for the following known characterizations of the two highest mixed Ehrhart coefficients (see [22] for the case k = d − 1 and [21, Lemma 3.7 and 3.8]). Corollary 2.5 (Steffens, Theobald). Let P = (P 1 , . . . , P k ) be lattice polytopes such that d = dim(P 1 + · · · + P k ). Then where the sum runs over all α ∈ Z k ≥1 with |α| = d. In particular, ME P (n) is a polynomial of degree exactly d.
Proof. By Theorem 2.4, me d (P) is the sum of the coefficients of E P (n 1 , . . . , n d ) of total degree d. These are coefficients of E d P (n 1 , . . . , n k ) = vol(n 1 P 1 + · · · + n k P k ) that are given by the mixed volumes. For the second claim, it is sufficient to note that since the sum P 1 + · · · + P k is full-dimensional, E d P is not identically zero and all coefficients are non-negative.
For k = d, the previous corollary recovers a result of Bernstein [4] (5) A similar reasoning also allows us to give an expression for the second highest coefficient me d−1 (P ). For an integral linear functional a ∈ (Z d ) ∨ , the rational subspace a ⊥ ⊆ R d is equipped with a lattice a ⊥ ∩Z d and a volume form vol a so that a fundamental parallelpiped has unit volume. We denote the face of a polytope P ⊆ R d where a is maximized by P a and consider it (after translation) as a polytope inside a ⊥ . Similarly, we write MV a (DMV a ) for the (discrete) mixed volume of a collection of polytopes in a ⊥ computed using vol a .
Corollary 2.6 (Steffens, Theobald). Let P = (P 1 , . . . , P k ) be a collection of d-dimensional lattice polytopes in R d . Then where a ranges over the primitive facet normals of P 1 + · · · + P k .
Proof. For a full-dimensional lattice polytope P ⊆ R d , the second highest Ehrhart coefficient can be expressed as where a ranges over all primitive vectors of facets of P (see, e.g., [2,Thm. 5.6]). This, of course, is a finite sum as P has only finitely many facets. It follows that (−1) k−|J| vol a (P a J ).
Since e d (P ) = vol d (P ) for a d-dimensional lattice polytope P ⊆ R d , (2) implies that the inner sum in the previous expression equals me d−1 (P a 1 , . . . , P a k ). The result now follows from Corollary 2.5 applied to P a = (P a 1 , . . . , P a k ). Observe that if a is a facet normal for P J , then a is a facet normal for P I for all I ⊇ J. Hence, the above sum is over all primitive facet normals of P 1 + · · · + P k .
Next, we will show that for the special case P 1 = P 2 = · · · = P k = P , the mixed Ehrhart polynomial can be expressed in terms of the h * -vector of P ; see Section 4. For now, let us mention that Stanley [18] showed that there exist non-negative integers h * i (P ) such that In terms of this presentation we get: Proposition 2.7. Let P ⊆ R d be a d-dimensional lattice polytope and let P = (P, . . . , P ) be a collection of k copies of P . Then In particular, Proof. From the definition of the mixed Ehrhart polynomial, we infer Using that E iP (n) = E P (in) and the expression of E P (n) as in (6), this yields This shows the first claim. For the second claim it suffices to observe that DMV(P) = ME P (1) and to check that, in this case, the inner sum in the above expression equals This is reminiscent to the relation between f -vectors and h-vectors in the enumerative theory of simplicial polytopes. Investigating this analogy further yields a theory of discrete mixed valuations; see [14].
Proof. By the hypothesis, P = P 1 +· · ·+P k is affinely isomorphic to the Cartesian product P 1 × · · · × P k . For any J ⊆ [k], this implies that P J is the intersection of P with the linear span of P J . And for I, J ⊆ [k], we see that the intersection P I ∩ P J is exactly P I∩J . Hence For the case that all polytopes are segments, the previous proposition can be used to recover (5).

Non-negativity of the discrete mixed volume and the mixed Ehrhart polynomial
Bihan [5] showed the following fundamental non-negativity result.
Thus the mixed Ehrhart polynomial evaluates to non-negative integers for all positive integers n.
For a proof of Theorem 3.1, Bihan develops a theory of irrational mixed decompositions which yields a technical induction on dimension. The discrete mixed volume is a particular combinatorial mixed valuation in the sense of [14]. It is shown in [14] that the discrete mixed volume is monotone with respect to inclusion, which implies Theorem 3.1. The proofs in [14] are less technical but set in the context of the polytope algebra. In this section we give simple and geometrically sound proofs for the special cases k ∈ {2, d−1, d} and P 1 = · · · = P k = P . For the case k = d this is a consequence of (5). For P 1 = · · · = P k = P this follows from Proposition 2.7 together with Stanley's result on the non-negativity of the h * -vector; see [18,15].
Direct proof of Theorem 3.1 for k = 2. Let P 1 and P 2 be lattice polytopes in R d . Since the mixed Ehrhart polynomial is invariant under translation of the polytopes, we may assume that 0 is a common vertex of P 1 and P 2 , P 1 ∩ P 2 = {0} and that there is a hyperplane H weakly separating P 1 from P 2 . Hence P 1 ∪ P 2 ⊆ P 1 + P 2 . It follows that for all n ∈ Z ≥0 , i.e., ME P 1 ,P 2 (n) ≥ 0 for all n ∈ Z ≥0 .
For k = d − 1, a direct proof idea has already been used in the framework of tropical geometry and d-dimensional Pick-type formulas in [22]. The following is a proof along similar lines.
Direct proof of Theorem 3.1 for k = d − 1. By Theorem 2.4, the mixed Ehrhart polynomial of a collection of lattice polytopes P = (P 1 , . . . , P d−1 ) ⊆ R d is of the form It suffices to show that me d (P) and me d−1 (P) are non-negative. Since the mixed volume is multilinear and symmetric in its entries, Corollary 2.5 yields For the remaining coefficient, Corollary 2.6 together with the fact that k = d − 1 yields where the sum is over all primitive facet normals of P 1 + · · · + P d−1 .

Mixed h * -polynomials
For a d-dimensional lattice polytope P , let Ehr P (z) = n≥0 E P (n)z n be the Ehrhart series of P . As E P (n) is a polynomial in n of degree d, there exist integers h * 0 (P ), . . . , h * d (P ) such that see, e.g., [1,13]. The polynomial in the numerator is called the h * -polynomial of P (also known as δ-polynomial [20]) and denoted by h * P (z). Similarly, the coefficient vector h * (P ) := (h * 0 (P ), . . . , h * d (P )) is called the h * -vector of P . Stated differently, the Ehrhart polynomial of a d-dimensional lattice polytope P can be expressed as in (6). By Stanley's non-negativity theorem [18], the coefficients h * 0 (P ), h * 1 (P ), . . . , h * d (P ) are non-negative integers. Moreover, it is known that where int(P ) denotes the interior of the polytope P . The definition of mixed Ehrhart polynomials prompts the notion of a mixed h * -vector h * (P) = (h * 0 (P), . . . , h * d (P)) of a collection P = (P 1 , . . . , P k ) of lattice polytopes, which is given by where d = dim(P 1 +· · ·+P k ). In this section, we study properties of h * (P) for a collection of full-dimensional lattice polytopes, i.e., dim P i = d for all i. Though Stanley's h * -nonnegativity does not extend to the mixed h * -vector as we will see below, we show that for large enough dilates rP := (rP 1 , . . . , rP k ), r 0, the corresponding mixed h *polynomial h * rP (z) = h * 0 (rP) + · · · + h * d (rP)z d has only real roots and hence the mixed h * -vector is log-concave. This is in line with results of Diaconis and Fulman [10] for the usual h * -vector.
Remark 4.1. Note that the contribution of the index set J = ∅ in the discrete mixed volume (1) is (−1) k . For the mixed Ehrhart series n≥0 ME P 1 ,...,P k (n)z n this induces for the index set J = ∅ the contribution Since we assume that all polytopes P i have the same dimension, linearity and Remark 4.1 allow to write the mixed h * -vector as (10) h The next lemma collects some elementary properties of mixed h * -vectors.
(ii) We know from (7) that for a d-dimensional lattice polytope P ⊆ R d . This fact combined with (10) yields where the last equality follows from (2). For the second claim, it suffices to note that DMV(P 1 , P 2 ) is non-negative (see Section 3).
A natural question when dealing with integer vectors is whether all entries are nonnegative. The previous lemma provides some positive results in this direction. However, as opposed to the h * -vector of a single lattice polytope, the next example shows that in general it is not true that the coefficients of the mixed h * -vector are non-negative. Though we have just seen that there do exist collections of polytopes with negative mixed h * -vector entries, observe that for m ≥ 2 all entries of h * (m∆ 3 , m∆ 3 ) = (0, m 3 + 2m, 4m 3 , m 3 − 2m 2 ) are indeed non-negative, and the leading coefficients in m are the Eulerian numbers (0, 1, 4, 1) (see below). We therefore propose to study the following question. In Corollary 4.6 we will show that asymptotically, i.e., if one considers high enough dilations of d-dimensional polytopes, the mixed h * -vector h * (P) always becomes non-negative. This suggests that it might be enough to require the lattice polytopes to contain "sufficiently many" interior points. Before we provide our main result of this section, we recall the definition of the d th Eulerian polynomial A d (z) = d k=1 A(d, k)z k . One, out of several, combinatorial approaches to define A d (z) is the following: It is known that the Eulerian polynomials A d (z) have only simple and real roots and all roots are negative. The following result, which is a generalization of Theorem 5.1 in [10], provides a relation between Eulerian polynomials and mixed h * -polynomials of lattice polytopes.
Proof. For a d-dimensional lattice polytope P ⊆ R d , the Ehrhart polynomial in the usual basis takes the form E P (n) = e d (P )n d + e d−1 (P )n d−1 + · · · + e 0 (P ) where e d (P ) = vol(P ). For the Ehrhart series we compute By comparing E rP (n) with E P (rn), we see that e i (rP ) = r i e i (P ) and hence Now from (10) together with the fact that rP ∅ = {0} for all r > 0, we infer which shows the claim.
For k = 1 this also yields the result of Diaconis and Fulman [10, Theorem 5.1] on the asymptotic behavior of the usual h * -polynomial of a lattice polytope. Similar results have also been achieved by Brenti and Welker [7] and Beck and Stapledon [3]. Before we provide some almost immediate consequences of the previous result, we recall that a sequence (a 0 , a 1 , . . . , a d ) of real numbers is called log-concave if a 2 i ≥ a i−1 a i+1 for all 1 ≤ i ≤ d − 1. The sequence (a 0 , a 1 , . . . , a d ) is called unimodal if there exists a 0 ≤ ≤ d such that a 0 ≤ · · · ≤ a ≥ · · · ≥ a d .
Proof. (i) First observe that, by Theorem 4.5, the roots of h * rP (z) converge to the roots of A d (z). Moreover, since the roots of A d (z) are known to be all distinct and negative, there exists a positive integer R such that for r ≥ R, h * rP (z) has only simple and real (negative) roots. Otherwise, since complex roots come up in pairs (if τ is a complex root, also its complex conjugateτ is a root), the Eulerian polynomial A d (z) would be forced to have a root of multiplicity 2, which yields a contradiction. (ii) Using (2) and the fact that e d (P ) = vol(P ) for a d-dimensional lattice polytope P ⊆ R d , we recognize the right-hand side in Theorem 4.5 as me d (P)A d (z). By Corollary 2.5, me d (P) is a non-negative linear combination of mixed volumes of P 1 , . . . , P k and thus positive. Since the coefficients of the Eulerian polynomials (besides the constant term which is 0) are all positive as well, Theorem 4.5 then implies that the sequence converges to a polynomial, whose coefficients, except for the constant term (which equals 0), are positive. Hence, there has to exist a positive integer R (depending on P) such that for r ≥ R all but the constant coefficient of h * rP (z) are positive. This shows (ii). (iii) Using the first two parts, we know that there exists a positive integer R such that for r ≥ R the polynomial h * rP (z) has, except for the constant term, positive coefficients and is real-rooted. Theorem 1.2.1 of [6] implies that h * (rP) is log-concave. Since by the choice of r, this sequence does not have internal zeros, it is unimodal by [6, Section 2.5].