The Arithmetic Circuit Combinatorial Nullstellensatz

The Arithmetic Circuit Combinatorial Nullstellensatz is NP-hard

Announce Type: new Abstract: A multivariate polynomial on $n$ variables $x_1,\ldots,x_n$ of total degree $n$ over $\mathbf{Z}_2$ containing the multilinear monomial $\prod_{i=1}^n x_i$ is by the combinatorial nullstellensatz Comput., 1999] known to always have a nonroot. We show that there cannot be a randomised polynomial time algorithm that given an arithmetic circuit of polynomial size formally computing such a polynomial, locates a nonroot with constant nonzero probability unless RP=NP.