This paper shows that, assuming $\mathsf{P}\ne\mathsf{NP}$, sparse univariate polynomial GCD cannot be computed in fixed-output polynomial time over any fixed computable field. Precisely, there is no deterministic algorithm that can compute the GCD of sparse univariate polynomials over any fixed computable field with total bit complexity polynomial in the input size, with the output size as a fixed parameter. In particular, there is no deterministic algorithm for sparse polynomial GCD that runs in polynomial time with respect to both the input and the output sizes. This settles the open problem posed as Challenge 5 in the sparsity challenges. Furthermore, we show that the Roots of Unity Detection problem over finite fields is NP-hard under random reductions; that is, determining whether the GCD of a sparse univariate polynomial and $x^n - 1$ has nonzero degree is NP-hard.

翻译：暂无翻译