We show that for every toric surface X apart from P2 and P1×P1 and every ample line bundle L on X there exists an ample polarisation A for X, such that the syzygy bundle ML⊗d associated to the tensor power L⊗d is not stable with respect to A for every d sufficiently large.

For any polyhedral norm, the bisector of two points is a polyhedral complex. We study combinatorial aspects of this complex. We investigate the sensitivity of the presence of labelled maximal cells in the bisector relative to the position of the two points. We thereby extend work of Criado, Joswig and Santos (2022) who showed that for the tropical distance function the presence of maximal cells is encoded by a polyhedral fan, the bisection fan. We initiate the study of bisection cones and bisection fans with respect to arbitrary polyhedral norms. In particular, we show that the bisection fan always exists for polyhedral norms in two dimensions. Furthermore, we determine the bisection fan of the & ell;1-norm and the & ell;infinity-norm as well as the discrete Wasserstein distance in arbitrary dimensions. Intricate combinatorial structures, such as the resonance arrangement, make their appearance. We apply our results to obtain bounds on the combinatorial complexity of the bisectors.

We generalize R. P. Stanley's celebrated theorem that the h⁎-polynomial of the Ehrhart series of a rational polytope has nonnegative coefficients and is monotone under containment of polytopes. We show that these results continue to hold for weighted Ehrhart series where lattice points are counted with polynomial weights, as long as the weights are homogeneous polynomials decomposable as sums of products of linear forms that are nonnegative on the polytope. We also show nonnegativity of the h⁎-polynomial as a real-valued function for a larger family of weights. We explore the case when the weight function is the square of a single (arbitrary) linear form. We show stronger results for two-dimensional convex lattice polygons and give concrete examples showing tightness of the hypotheses. As an application, we construct a counterexample to a conjecture by Berg, Jochemko, and Silverstein on Ehrhart tensor polynomials.

The Ehrhart polynomial ehrP(n) of a lattice polytope P gives the number of integer lattice points in the n-th dilate of P for all integers n ≥ 0. The degree of P is defined as the degree of its h∗-polynomial, a particular transformation of the Ehrhart polynomial with many useful properties which serves as an important tool for classification questions in Ehrhart theory. A zonotope is the Minkowski (pointwise) sum of line segments. We classify all Ehrhart polynomials of lattice zonotopes of degree 2 thereby complementing results of Scott (Bull Aust Math Soc 15(3), 395–399, 1976), Treutlein (J Combin Theory Ser A 117(3), 354–360, 2010), and Henk and Tagami (Eur J Combin 30(1), 70–83, 2009). Our proof is constructive: by considering solid-angles and the lattice width, we provide a characterization of all 3-dimensional zonotopes of degree 2.

We consider the task of reconstructing polytopes with fixed facet directions from finitely many support function evaluations. We show that for a fixed simplicial normal fan, the least-squares estimate is given by a convex quadratic program. We study the geometry of the solution set and give a combinatorial characterization for the uniqueness of the reconstruction in this case. We provide an algorithm that, under mild assumptions, converges to the unknown input shape as the number of noisy support function evaluations increases. We also discuss limitations of our results if the restriction on the normal fan is removed.

Over a decade ago De Loera, Haws and Koppe conjectured that Ehrhart polynomials of matroid polytopes have only positive coefficients and that the coefficients of thecorresponding h*-polynomials form a unimodal sequence. The first of these intensively studied conjectures has recently been disproved by the first author who gave counterexamples in all ranks greater than or equal to three. In this article we complete the picture by showing that Ehrhart polynomials of matroids of lower rank have indeed only positive coefficients. Moreover, we show that they are coefficient-wise bounded by the Ehrhart polynomials of minimal and uniform matroids. We furthermore address the second conjecture by proving that h*-polynomials of matroid polytopes of sparse paving matroids of rank two are real-rooted and therefore have logconcave and unimodal coefficients. In particular, this shows that the h*-polynomial of the second hypersimplex is realrooted, thereby strengthening a result of De Loera, Haws and Koppe.

We characterize all signed Minkowski sums that define generalized permutahedra, extending results of Ardila-Benedetti-Doker (Discrete Comput. Geom. 43 (2010), no. 4, 841-854). We use this characterization to give a complete classification of all positive, translation-invariant, symmetric Minkowski linear functionals on generalized permutahedra. We show that they form a simplicial cone and explicitly describe their generators. We apply our results to prove that the linear coefficients of Ehrhart polynomials of generalized permutahedra, which include matroid polytopes, are non-negative, verifying conjectures of De Loera-Haws-Koppe (Discrete Comput. Geom. 42 (2009), no. 4, 670-702) and Castillo-Liu (Discrete Comput. Geom. 60 (2018), no. 4, 885-908) in this case. We also apply this technique to give an example of a solid-angle polynomial of a generalized permutahedron that has negative linear term and obtain inequalities for beta invariants of contractions of matroids.

We consider the integer point transform $$\sigma _P (\mathbf {x}) = \sum _{\mathbf {m}\in P\cap \mathbb {Z}^n} \mathbf {x}^\mathbf {m}\in \mathbb {C}[x_1^{\pm 1},\ldots, x_n^{\pm 1}]$$σP(x)=∑m∈P∩Znxm∈C[x1±1,…,xn±1] of a polytope P⊂ Rn. We show that if P is a lattice polytope then for any polytope Q the sequence {σkP+Q(x)}k≥0 satisfies a multivariate linear recursion that only depends on the vertices of P. We recover Brion’s Theorem and by applying our results to Schur polynomials we disprove a conjecture of Alexandersson (2014).

We study rational generating functions of sequences {an } n≥0 that agree with a polynomial and investigate symmetric decompositions of the numerator polynomial for subsequences {arn } n≥0. We prove that if the numerator polynomial for {an } n≥0 is of degree s and its coefficients satisfy a set of natural linear inequalities, then the symmetric decomposition of the numerator for {arn } n≥0 is real-rooted whenever r = max{s,d+1-s}.Moreover, if the numerator polynomial for {an } n≥0 is symmetric, then we show that the symmetric decomposition for {arn } n≥0 is interlacing.We apply our results to Ehrhart series of lattice polytopes. In particular, we obtain that the h*-polynomial of every dilation of a d-dimensional lattice polytope of degree s has a real-rooted symmetric decomposition whenever the dilation factor r satisfies r = max{s,d + 1 - s}. Moreover, if the polytope is Gorenstein, then this decomposition is interlacing. 

Eulerian polynomials are fundamental in combinatorics and algebra. In this paper we study the linear transformation A : R[t] -> R[t] defined by A(t(n)) = A(n)(t), where A(n)(t) denotes the n-th Eulerian polynomial. We give combinatorial, topological and Ehrhart theoretic interpretations of the operator A, and investigate questions of unimodality and real-rootedness. In particular, we disprove a conjecture by Brenti (1989) concerning the preservation of real zeros, and generalize and strengthen recent results of Haglund and Zhang (2019) on binomial Eulerian polynomials.