Weighted counting of solutions to sparse systems of equations.

Given complex numbers $w_1, ldots, w_n$, we define the weight $w(X)$ of a set $X$ of 0-1 vectors as the sum of $w_1^{x_1} cdots w_n^{x_n}$ over all vectors $(x_1, ldots, x_n)$ in $X$. We present an algorithm, which for a set $X$ defined by a system of homogeneous linear equations with at most $r$ variables per equation and at most $c$ equations per variable, computes $w(X)$ within relative error $epsilon u003e0$ in $(rc)^{O(ln n-ln epsilon)}$ time provided $|w_j| leq beta (r sqrt{c})^{-1}$ for an absolute constant $beta u003e0$ and all $j=1, ldots, n$. A similar algorithm is constructed for computing the weight of a linear code over ${Bbb F}_p$. Applications include counting weighted perfect matchings in hypergraphs, counting weighted graph homomorphisms, computing weight enumerators of linear codes with sparse code generating matrices, and computing the partition functions of the ferromagnetic Potts model at low temperatures and of the hard-core model at high fugacity on biregular bipartite graphs.