We consider the following computational problem: we are given two coprime univariate polynomials f ₀ and f ₁ over a ring 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 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 ).