Slice and Partition Rank Criteria for Polynomial Zero-Avoidance
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 Erd\H{o}s--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 Erd\H{o}s--Ginzburg--Ziv problem over $\mathbb{F}_5^n$ associated with the fourth elementary symmetric polynomial.