# Algebraic Unimodular Counting

## ABSTRACT

Given an $n \times m$ unimodular matrix $A$ and an integral right hand side $b$. We consider the counting function $\phi_A(b)= | {x : Ax=b, x \geq 0 } |$. The functions $\phi_A(b)$ appear in many different areas including Representation Theory (Kostant partition functions) and Statistics (Contingency tables). It is known that $\phi_A(b)$ is a piecewise polynomial function in terms of the parameters $b$ and the degree of the polynomials components is $m-n$. The domains of valid polynomiality form the well-known chamber complex appearing in the theory of secondary polytopes. Recently, $\phi_A(b)$ has also been representated as a generating function based on the work on Barvinok and Brion. In this talk we will discuss 1) The different algebraic representations of the counting function $\phi_A(b)$ and how they help to effectively compute concrete answers and derive explicit combinatorial formulas. 2) The geometric structure of the chamber complex and its chambers. This is joint work with Bernd Sturmfels (UC Berkeley).

Speaker's Contact Info: deloera(at-sign)math.ucdavis.edu