Algebraic Unimodular Counting
Jesús De Loera
U.C. Davis
March 21,
4:15pm
refreshments at 3:45pm
2338
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 $mn$. The
domains of valid polynomiality form the wellknown 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(atsign)math.ucdavis.edu
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