Polynomial GCDs by Syzygies

Abstract

We provide a simple method, using Gröbner bases over modules, to compute multivariate polynomial greatest common divisors. The approach we show is flexible, adaptable to algebraic extensions of the rationals or prime fields, and is notably faster than prior methods that work with Gröbner bases. It can be used in situations where sparse interpolation might be difficult to implement, e.g. when there are few points for interpolation (small prime fields) or in the presence of non-numeric algebraic relations.