Solvability of a System of Bivariate Polynomial Equations over a Finite Field

Abstract

We investigate the complexity of the following polynomial solvability problem: Given a finite field ${\mathbb F}_{q}$ and a set of polynomials

$$f_{1}(x,y),f_{2}(x,y),...,f_{n}(x,y),g(x,y) \ \epsilon \ {\mathbb F}_{q} [x,y]$$

determine the ${\mathbb F}_{q}$-solvability of the system

$$f_{1}(x,y)=f_{2}(x,y)=...=f_{n}(x,y)=0 \ {\rm and} \ {\it g}(x,y) \neq 0$$

We give a deterministic polynomial-time algorithm for this problem.