Related papers: A finite calculus approach to Ehrhart polynomials
We determine lattice polytopes of smallest volume with a given number of interior lattice points. We show that the Ehrhart polynomials of those with one interior lattice point have largest roots with norm of order n^2, where n is the…
It is believed arXiv:0808.2762, arXiv:math/9904055 that, among the coefficients entering Kontsevich's formality quasi-isomorphism arXiv:q-alg/9709040, there are irrational (possibly even transcendental) numbers. In this paper, we prove that…
We prove a quantitative Roth-type theorem for polynomial corners in $\mathbb{R}^2$. Let $P_1$ and $P_2$ be two linearly independent polynomials with zero constant term. We show that any measurable subset of $[0,1]^2$ with positive measure…
A $3$-dimensional polytope $P$ is $k$-equiprojective when the projection of $P$ along any line that is not parallel to a facet of $P$ is a polygon with $k$ vertices. In 1968, Geoffrey Shephard asked for a description of all equiprojective…
We show that the Ehrhart h-vector of an integer Gorenstein polytope with a regular unimodular triangulation satisfies McMullen's g-theorem; in particular, it is unimodal. This result generalizes a recent theorem of Athanasiadis (conjectured…
We study properties of convex hulls of (co)adjoint orbits of compact groups, with applications to invariant theory and tensor product decompositions. The notion of partial convex hulls is introduced and applied to define two numerical…
We present a structural resolution to the exact evaluation of the partition function $p_k(n)$, systematically overcoming the limitations of traditional recursive and asymptotic methods. By framing the partition polytope $\mathcal{P}_{n,k}$…
In this paper we use the technique of Hopf algebras and quasi-symmetric functions to study the combinatorial polytopes. Consider the free abelian group $\mathcal{P}$ generated by all combinatorial polytopes. There are two natural bilinear…
Consider a polynomial $f$ with a convenient Newton polytope $P$ and generic complex coefficients. By the global version of the Kouchnirenko formula, the hypersurface $\{f = 0\} \subset \mathbb{C}^n$ has the homotopy type of a bouquet of…
We investigate the Ehrhart polynomial for the class of 0-symmetric convex lattice polytopes in Euclidean $n$-space $\mathbb{R}^n$. It turns out that the roots of the Ehrhart polynomial and Minkowski's successive minima are closely related…
The n'th Birkhoff polytope is the set of all doubly stochastic n-by-n matrices, that is, those matrices with nonnegative real coefficients in which every row and column sums to one. A wide open problem concerns the volumes of these…
Recent work has focused on the roots z of the Ehrhart polynomial of a lattice polytope P. The case when Re(z) = -1/2 is of particular interest: these polytopes satisfy Golyshev's "canonical line hypothesis". We characterise such polytopes…
We give a local Euler-Maclaurin formula for rational convex polytopes in a rational euclidean space . For every affine rational polyhedral cone C in a rational euclidean space W, we construct a differential operator of infinite order D(C)…
The purpose of this paper is the formal verification of a counterexample of Santos et al. to the so-called Hirsch Conjecture on the diameter of polytopes (bounded convex polyhedra). In contrast with the pen-and-paper proof, our approach is…
A lattice polytope is "free" (or "empty") if its vertices are the only lattice points it contains. In the context of valuation theory, Klain (1999) proposed to study the functions $\alpha_i(P;n)$ that count the number of free polytopes in…
It is shown that no purely topological condition implies the equality of the polynomial and rational hulls of a set: For any compact subset $K$ of a Euclidean space, there exists a set $X$, in some ${\mathbb C}^N$, that is homeomorphic to…
This note is a comment to the paper by D.R.Heath-Brown and B.Z.Moroz (Math Proc. Camb. Phil. Soc. 125 (1999)). That paper concerns with the projective surface $S$ in $\mathbb{P}^{3}$ defined by the equation $x_{1}x_{2}x_{3}=x_{4}^{3}$. It…
McMullen's formulas or local formulas for Ehrhart coefficients are functions on rational cones that determine the $i$-th coefficient of the Ehrhart polynomial as a weighted sum of the volumes of the i-dimensional faces of a polytope. This…
Ehrhart theory is the study of the enumeration of lattice points in lattice polytopes. Equivariant Ehrhart theory is a generalization of Ehrhart theory that takes into account the action of a finite group acting via affine transformations…
We study a class of polynomials that has all of its roots on the critical line and shares many properties with Ehrhart polynomials. Braun showed that the roots of Ehrhart polynomials are bounded quadratically and Higashitani provided…