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 [Alon, Comb. Probab. 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. The result holds even when the individual degree of every variable in the input polynomial is at most two.

翻译：关于$n$个变量$x_1,\ldots,x_n$、总次数为$n$且在$\mathbf{Z}_2$上的多元多项式，若其包含多重线性单项式$\prod_{i=1}^n x_i$，则根据组合零点定理[Alon, Comb. Probab. Comput., 1999]，该多项式总存在一个非根。我们证明，不存在一个随机多项式时间算法，能够在给定一个多项式大小的算术电路（该电路形式化地计算此类多项式）时，以非零常数概率定位一个非根，除非RP=NP。该结论即使输入多项式中每个变量的个体次数至多为二时仍然成立。