Approximate polynomial [formula omitted]: 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 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 R = 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 R = Z ). Our work adds to the existing notions of “approximate gcd”.