Tan Nhat Tran (Hokkaido University)


Title: Quasi-polynomials in arrangement and Ehrhart theories

Abstract: We are going to investigate a typical problem in enumerative combinatorics: counting the sizes of the sets depending upon a positive integer $q$. The result often is polynomials (e.g., the chromatic polynomial of a graph), and sometimes quasi-polynomials. Generally speaking, a quasi-polynomial is a generalization of polynomials, of which the coefficients may not come from a ring but instead are periodic functions with integral period. One of the most classical examples is the Ehrhart quasi-polynomial that counts the number of integral points in the $q$-fold dilation of a rational polytope. In the arrangement theory, a quasi-polynomial appears when we count the size of the complement of an integral hyperplane arrangement modulo $q$ - the characteristic quasi-polynomial due to Kamiya-Takemura-Terao. In the first part, we present an interpretation of the constituents of the characteristic quasi-polynomial through toric arrangements. In the second part, we show a link between the characteristic and Ehrhart quasi-polynomials in connection with root systems and Eulerian polynomials. The first part is based on a joint work with Yoshinaga (Hokkaido), the second part is based on a joint work with Ashraf (Western Ontario) and Yoshinaga.