🤖 AI Summary
This work investigates the computational complexity of finding a non-vanishing point for a polynomial given by an arithmetic circuit, with a focus on the constructive aspects under the combinatorial Nullstellensatz framework. By leveraging tools from computational complexity theory, arithmetic circuit models, and probabilistic reductions, the study establishes—for the first time—that the problem remains NP-hard even when the individual degree of every variable is at most two. Consequently, under the standard complexity assumption that RP ≠ NP, no randomized polynomial-time algorithm can find such a non-zero evaluation point with constant success probability. This result underscores the inherent difficulty of constructively realizing the combinatorial Nullstellensatz, revealing fundamental limitations in efficiently producing witnesses guaranteed to exist by this algebraic principle.
📝 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 [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.