It is important to solve linear systems $Ax=b$, where the matrix $A$ has integral coefficients, and $x$ is a vector with integral positive coefficients. The number of solutions is a function $P_A(b)$ of $b$, called the partition function: if\[A = \begin{bmatrix}1 & 1 & 1\end{bmatrix}\] the equation reads $x_1+x_2+x_3=b$, and the number of solutions is $P_A(b)=\frac{1}{5!}(b+1)(b+2)(b+3)$ so represents the number of ways to partition a positive integer $b$ in 3 parts, adding up to $b$.

Many examples of such equations arise in statistics, algebraic geometry, representation theory, etc. The function $P_A(b)$ is “locally quasi polynomial”, a result due to Ehrhart, and which is based on the relation between the number $P_A(b)$ and the Riemann-Roch formula on a toric variety. I will present here a more efficient approach based on the cohomology of the complement of hyperplanes. Due to a multidimensional residue formula, this number $P_A(b)$ may be calculated as an integral over a cycle $C$ in the complement.

This is joint work with Velleda Baldoni-Silva and Andras Szenes.