Approximate polynomial gcd: Small degree and small height perturbations

We consider the following computational problem: we are given two coprime univariate polynomials f(0) and f(1) over a ring D 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 D = Z). Our work adds to the existing notions of "approximate gcd". (C) 2010 Elsevier Ltd. All rights reserved.