Overdetermined Systems of Sparse Polynomial Equations

Abstract

In this paper, we show that for a system of univariate polynomials given in sparse encoding we can compute a single polynomial defining the same zero set in quasilinear time in the logarithm of the degree. In particular, it is possible to decide whether such a system of polynomials has a zero in quasilinear time in the logarithm of the degree. The underlying algorithm relies on a result of Bombieri and Zannier on multiplicatively dependent points in subvarieties of an algebraic torus. We also present the following conditional partial extension to the higher-dimensional setting. Assume that the effective Zilber conjecture holds. Then, for a system of multivariate polynomials given in sparse encoding we can compute a finite collection of complete intersections outside hypersurfaces that defines the same zero set in quasilinear time in the logarithm of the degree.