Approximate Polynomial gcd: Small Degree and Small Height Perturbations

Abstract

We consider the following computational problem: we are given two coprime univariate polynomials f ₀ and f ₁ over a ring $\mathcal{R}$ and want to find whether after a small perturbation we can achieve a large gcd. We solve this problem in polynomial time for two notions of “large” (and “small”): large degree (when $\mathcal{R} = \mathbb{F}$ is an arbitrary field, in the generic case when f ₀ and f ₁ have a so-called normal degree sequence), and large height (when $\mathcal{R} =\mathbb{Z}$).