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2. Ehrhart h^\ast-polynomial: Unimodality and Real-Rootedness


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      Problem 2.02.

      Does the Pitman-Stanley polytope have real-rooted h^*-polynomial?
        1. Remark. [Alejandro Morales] In unpublished work of (Benedetti-Gonzalez D’Leon-Hanusa-Harris) the h^* polynomial of the Pitman-Stanley polytope of dimension n is h^*(z)=\sum_{\pi \in PF_n} z^{des(\pi)}, where PF_n is the set of parking functions of size n and des(\pi) is the number of descents of \pi which are the indices i such that \pi(i)\geq \pi(i+1).


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              Problem 2.04.

              Study the h^*-polynomials of (not necessarily convex) 3-polytopes.


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                  Problem 2.06.

                  For a lattice polytope P, let P^n denote the Cartesian product of P with itself n times (n\in \N). Investigate what happens to the h^*-polynomial of P^n. Does it eventually become real-rooted?


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                      Problem 2.08.

                      For a symmetric edge polytope, can we obtain regular unimodular triangulations using liftings with low heights? If so, how many such triangulations can be obtained?


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                          Problem 2.1.

                          What are the ways of “measuring” root polytopes, i.e., edge polytopes?

                          Furthermore, how should Grothendieck polynomials be specialized to obtain Ehrhart h^*-polynomials?


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                              Problem 2.12.

                              Consider the order polynomial, denoted \Omega(P,t), of a finite poset P. Let n be the number of elements in P.

                              Conjecture [Kahn, Saks] \frac{\Omega(P,t)}{t^n} is nonincreasing for all t\in \Z_{>0}.

                              This statement is known to be true if all coefficients of \Omega(P,t) are positive. Investigate the conjecture more generally.

                              (See solution to Exercise 3.163(b) in Enumerative Combinatorics Vol. 1 by Stanley).

                                •     Definition: Let P be a lattice polytope of dimension d. Let \omega\in \Q[x_1,\ldots,x_d] be a homogeneous polynomial. We define the \omega-Ehrhart function i(P;\omega;n) := \sum_{\alpha\in nP\cap \Z^d} \omega(\alpha).

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                                  Problem 2.14.

                                  Note that i(P;1;n) = i(P;n). It is known that i(P;\omega;n) is a polynomial with leading coefficient given by \int_P \omega. Investigate \omega-Ehrhart function for well-known classes of polytopes.


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                                      Problem 2.16.

                                      Identify polytopes with extremal h^*-vectors. That is, for a family of polytopes \mathcal{P} and index j, find P\in \mathcal{P} that minimize or maximize h_j^*. Potential families of interest include symmetric edge polytopes for a fixed degree sequence and simplices in Hermite normal form with fixed diagonal.


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                                          Problem 2.18.

                                          What relationship, if any, is there between heights of Hilbert basis elements and differences in h^*-coefficients? For example, consider h^*_{j+1} - h^*_j or more generally h^*_j - h^*_i.


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                                              Problem 2.2.

                                              Take the nonnegative hull of h^*-vectors of polytopes in a fixed dimension. Does this form a polyhedral cone?


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                                                  Problem 2.22.

                                                  Are there “holes” in the set of h^*-vectors of polytopes in a fixed dimension which are not a result of failing known linear inequalities?


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                                                      Problem 2.24.

                                                      How many facets do symmetric edge polytopes have?


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                                                          Problem 2.26.

                                                          Consider the convex hull of parking functions. Does this admit a unimodular triangulation? Furthermore, can the Ehrhart polynomial be obtained from this?


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                                                              Problem 2.28.

                                                              1. (a) Find 0/1-polytopes with non-unimodal, non-log-concave, etc. h^*-vectors.
                                                              2. (b) Find 0/1-polytopes with non-unimodal, non-log-concave, etc. f-vectors.
                                                              3. (c) Find 0/\pm1-polytopes with non-unimodal, non-log-concave, etc. h^*-vectors.


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                                                                  Problem 2.3.

                                                                  Determine a Veronese construction for rational polytopes.
                                                                    1. Remark. [Mariel Supina] I want to clarify what I meant with this problem. For lattice polytopes, we can apply a certain transformation to the h^*-polynomial of P to get the h^*-polynomial of kP for a positive integer k (see e.g. https://arxiv.org/pdf/0712.2645.pdf). Does such a transformation exist for rational polytopes? Does such a transformation exist in the setting of equivariant Ehrhart theory?


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                                                                          Problem 2.32.

                                                                          How many chambers are there for transportation polytopes for n\times n matrices? For example, how many Ehrhart polynomials are needed?


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                                                                              Problem 2.34.

                                                                              Identify classes of simplices with unimodal box polynomials.


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                                                                                  Problem 2.36.

                                                                                  Investigate a,b-decompositions for “nice” h^*-polynomials.


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                                                                                      Problem 2.38.

                                                                                      Investigate Stapledon’s Effectiveness Conjecture.

                                                                                      Conjecture. [Stapledon] Let P be a lattice polytope invariant under the action of a group G. Then the characters of the equivariant H^*-series is effective if and only if the equivariant H^*-series is a polynomial.

                                                                                      Stapledon proved that H^* effective implies H^* is a polynomial. The converse remains open.


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                                                                                          Problem 2.4.

                                                                                          A spinal graph G is a connected digraph on vertex set V=\{1,2,...,n\} with edge multiset E, where all edges are directed from smaller to larger vertices and E contains all edges of the form (i,i+1) for i=1,\ldots,n-1 (i.e., the “spine” of G).

                                                                                          What can be said about the h^*-polynomials of flow polytopes of spinal graphs?

                                                                                          Conjecture: descent polynomial of lower G-permutations \leq h^*(\mathcal{F}_1;z) \leq descent polynomial of upper G-permutations, where \leq indicates domination.

                                                                                              Cite this as: AimPL: Ehrhart polynomials: inequalities and extremal constructions, available at http://aimpl.org/ehrhartineq.