The Differential Counting Polynomial

Differential Polynomial Rings of Triangular Matrix Rings

Counting Stable Solutions of Sparse Polynomial Systems

Counting Affine Roots of Polynomial Systems

Counting Solutions to Polynomial Systems via Reductions

This paper provides both positive and negative results for counting solutions to systems of polynomial equations over a finite field. The general idea is to try to reduce the problem to counting solutions to a single polynomial, where the task is easier. In both cases, simple methods are utilized that we expect will have wider applicability (far beyond algebra). First, we give an efficient dete...

Counting Magic Squares in Quasi-polynomial Time

We present a randomized algorithm, which, given positive integers n and t and a real number 0 < < 1, computes the number |Σ(n, t)| of n× n non-negative integer matrices (magic squares) with the row and column sums equal to t within relative error . The computational complexity of the algorithm is polynomial in −1 and quasi-polynomial in N = nt, that is, of the order N log N . A simplified versi...