Hitting Sets for Polynomials with Small Partial Derivative Spaces

2026-09-28

AI Summary

This study addresses the problem of constructing efficient explicit hitting sets for derandomizing polynomials with bounded partial derivative spaces, particularly depth-3 powering circuits. The proposed methodology integrates algebraic geometry with combinatorial design theory, introducing Moura formal derivatives into this setting for the first time. The derivation leverages properties of Wronskian determinants combined with large language model-assisted proof techniques. As a result, this work constructs an explicit hitting set of size poly(n,d,r), effectively resolving the derandomization problem for this specific polynomial class. Furthermore, it provides a fully self-contained elementary proof and demonstrates the potential of AI-assisted mathematical discovery.

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.

Problem

• Hitting Set
• Polynomial Identity Testing
• Partial Derivative Space
• Depth-3 Powering Circuits
• Algebraic Complexity

Innovation

• Hitting Sets
• Partial Derivative Space
• Depth-3 Powering Circuits
• Formal Derivation
• Wronskian