Loading Web-Font TeX/Math/Italic
| Register
\newcommand{\Cat}{{\rm Cat}} \newcommand{\A}{\mathcal A} \newcommand{\freestar}{ \framebox[7pt]{$\star$} }

1. Ehrhart Positivity


    1. Add remark

      Conjecture 1.05.

      Let [n]:=\{1,\ldots,n\} be a groundset, and let i(P; t) denote the Ehrhart polynomial of a lattice polytope P. Then coefficient-wise, i(T_{r,n}; t) \leq i(\mathsf{Pan}_{s,r,n};t) \leq i(U_{r,n};t),
      where T_{r,n} is the rank r minimal matroid on [n], U_{r,n} is the rank r uniform matroid on [n], and \mathsf{Pan}_{s,r,n} is the rank r Panhandle matroid on [n].


        • Add remark

          Problem 1.1.

          What is known (or can be shown) about the Ehrhart positivity of polytopes related to Birkhoff polytopes (e.g., transportation polytopes)?


            • Add remark

              Problem 1.15.

              Are d-dimensional smooth polytopes with \leq d+3 facets Ehrhart positive or satisfy the higher integrality condition?


                • Add remark

                  Problem 1.2.

                  1. (a) Is there a closed formula for the Ehrhart polynomial of order polytopes of fence posets? Further, are these Ehrhart polynomials Ehrhart positive?
                  2. (b) Is the h^\ast-polynomial of the order polytopes from (a) real-rooted?
                  3. (c) Can this be extended to generalized permutahedra?


                    • Add remark

                      Problem 1.25.

                      Does a rational polygon P\subseteq \R^2 with quasi-period 1 satisfy Scott’s inequality? That is, assuming i\geq 1, is the inequality b\leq 2i+7 satisfied, where b and i represent the number of boundary and interior lattice points of P, respectively.


                        • Add remark

                          Problem 1.3.

                          Let ASM_n denote the nth alternating sign matrix polytope (defined as the convex hull in \mathbb{R}^{n^2} of the n\times n alternating sign matrices). It is known that i(ASM_n;t) is Ehrhart positive. What further can be said about i(ASM_n;t)? Furthermore, what can be said about h^*(ASM_n;z)?

                            •     Definition: Given two lattice polytopes P\subseteq \R^d and P'\subseteq\R^{d'}, the join of P and P' is given by P*P' := \mathrm{conv}\{(P,0_{d'},1) \cup (0_{d},P',0)\}.

                              Add remark

                              Problem 1.35.

                              For lattice polytopes P and P', which Ehrhart-theoretic properties pass through their join? For example, if P and P' are both Ehrhart positive, is P*P' as well?


                                • Add remark

                                  Problem 1.4.

                                  In types B, C, and D, the stretched Clebsch-Gordon coefficients are period-2 quasi-polynomials. Are the coefficients of the constituent polynomials always positive?


                                    • Add remark

                                      Problem 1.45.

                                      Consider rational polygons of denominator 2. The Ehrhart quasi-polynomial has two constituent polynomials, one of which is an Ehrhart polynomial. Classify the other constituent polynomial.

                                      Note that this is settled for up to 2 interior lattice points.


                                        • Add remark

                                          Problem 1.5.

                                          Characterize Ehrhart polynomials of tropical polytopes.


                                            • Add remark

                                              Problem 1.55.

                                              Study the Ehrhart polynomial of Harmonic polytopes and/or bipermutahedra.


                                                • Add remark

                                                  Problem 1.6.

                                                  1. (a) Consider the Ehrhart polynomial of a lattice polytope P. Below are two well-known representations of i(P;t): i(P;t) = \sum_{i=1}^d c_it^i = \sum_{i=0}^d h_i^* \binom{t+d-i}{d}.
                                                    Fix k\in \Z_{>0} and write i(P;t) in the basis \{(t-k)^i\}_i. (For example, if \deg(h^*(P;z)) = s, then i(P;t) is nonnegative in the basis \{(t-s+1)^i\}_{i=0}^{d}.)

                                                    Investigate what happens when changing k from 0 to 1 for non-Ehrhart positive polytopes (or other k).
                                                  2. (b) If P is Ehrhart positive in the standard basis, is it Ehrhart positive in the basis \{t^i(t+1)^{d-i}\}?

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