EAPSI: Unimodality of a polynomial associated to certain polytopes

Solus Liam T, Brentwood TN

Investigators

Abstract

High-dimensional polygons, commonly called polytopes, are geometric objects arising from finite sets. For certain types of polytopes we can produce a polynomial, called the h*-polynomial, that encodes interesting information about the polytope, such as its volume. One interesting property of the h*-polynomial, called unimodality, is when its coefficients rise and then fall when read from left-to-right. This project aims to show that the h*-polynomials of a well-studied collection of polytopes are unimodal. These polytopes are important structures in algebra, integer programming and other areas of mathematics. By proving the h*-polynomials of these fundamental polytopes are unimodal, this project intends to add to our understanding of both the collection of polytopes themselves, and the property of unimodality for h*-polynomials in general. This research will be conducted in collaboration with Dr. Takayuki Hibi, a leader in the study of h*-polynomials, at Osaka University in Japan. This project proposes a new geometric method to investigate unimodality of the h*-polynomial of the n,k-hypersimplex. Specifically, one can show that a well-studied unimodular triangulation of the n,k-hypersimplex restricts to a triangulation of a nested class of subpolytopes within the hypersimplex. It is well-known that a shellable unimodular triangulation can be used to compute the h*-polynomial of a lattice polytope. This project aims to define a shelling of this triangulation that respects the nesting of these subpolytopes. Through the analysis of this shelling the PI will study relations among the h*-polynomials of the subpolytopes and the hypersimplex. Particularly, the PI aims to relate the h*-polynomials of the subpolytopes in such a way that yields a result on unimodality of the h*-polynomial of the n,k-hypersimplex. This NSF EAPSI award is funded in collaboration with the Japan Society for the Promotion of Science.