Slice and Partition Rank Criteria for Polynomial Zero-Avoidance

Abstract

We study polynomial zero-avoidance over finite vector spaces by means of slice rank and partition rank. We first make the support-entropy method effective by showing how a finite dual certificate yields an explicit entropy gap whenever the coefficient support admits no probability distribution with uniform marginals. For the quadratic elementary symmetric polynomial over fields of characteristic three, the ternary structure of the coefficient support gives a certificate with optimal normalized margin and a uniform analytic bound for the corresponding higher-degree Erdős--Ginzburg--Ziv constant, avoiding a separate optimization for each field. We then use partition rank to handle solutions in pairwise distinct variables. Equality profiles are encoded by contracted local tensors, reducing the global problem to finitely many slice-rank estimates. Applying this reduction on the multiplicative torus gives restricted-alphabet zero-sum bounds with exponential base below the alphabet size. Coordinatewise inversion and support stratification then yield, to the best of our knowledge, the first nontrivial exponential bound for the higher-degree Erdős--Ginzburg--Ziv problem over $\mathbb{F}_5^n$ associated with the fourth elementary symmetric polynomial.

Disclosure

“ction gives only the trivial base 3, and, to the best of our knowledge, it remains open whether there exists c < 3 such that EGZ(3, Fn3 , 2) ≤ cn+o(n) . Declaration of generative AI assistance During the preparation of this work, the authors used OpenAI’s ChatGPT and Anthropic’s Claude as interactive aids in exploring possible approaches to mathematical and expository questions arising during the development of the”

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2026-08-23T19:15:54.612466Z

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