2008/01/01 by Benjamin Braun, Mike Develin · 1 citation
Mathematics · Psychology · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Algebraic structures and combinatorial models #Mathematics #Negativity effect #Polynomial #Properties of polynomial roots #Combinatorics #Discrete mathematics #Matrix polynomial #Mathematical analysis #Cognitive psychology #Psychology
paper · doi:10.1090/conm/452/08773
openalex publication_date 2008/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Stanley's non-negativity theorem is at the heart of many of the results in Ehrhart theory.In this paper, we analyze the root behavior of general polynomials satisfying the conditions of Stanley's theorem and compare this to the known root behavior of Ehrhart polynomials.We provide a possible counterexample to a conjecture of the second author, M. Beck, J. De Loera, J. Pfeifle, and R. Stanley, and contribute some experimental data as well.Let P be a convex polytope in R n with vertices in Z n and affine span of dimension d.We will refer to such polytopes as lattice polytopes and to elements of Z n as lattice points.By a remarkable theorem due to E. Ehrhart, [5], the number of lattice points in the t th dilate of P , for non-negative integers t, is given by a polynomial in t of degree d called the Ehrhart polynomial of P .In this paper we will investigate some differences between the root behavior of Ehrhart polynomials for elements of an arbitrary collection of polytopes of dimension less than or equal to d and the root behavior of an arbitrary collection of polynomials of degree less than or equal to d satisfying a certain non-negativity condition.This is easily seen by expanding the rational function as a formal power series.