Hitting Sets for Polynomials with Small Partial Derivative Spaces

Abstract

We give an explicit hitting set of size $\text{poly}(n,d,r)$ for the class of $n$-variate degree-$d$ polynomials whose partial derivative space is bounded by $r$, over any field $\mathbb{F}$ of characteristic zero. In particular, this yields a polynomial sized hitting set for the class of depth-$3$ powering circuits. The main technical insight is the construction of a "formal derivation'' and properties of the associated Wronskian with respect to this derivation, which was previously studied by Moura [Moura_2004] in a very different context. The proofs in this paper are elementary and completely self-contained. AI disclosure: The proof of this result was obtained during conversations [astra_proof] with OpenAI GPT-6 Astra. The proof presented in this writeup is a rewriting (in the authors' words) of the proof obtained by the AI model in a form that we believe is understandable to researchers.