On Vanishing Properties of Polynomials on Symmetric Sets of the Boolean Cube, in Positive Characteristic

TL;DR

This paper characterizes the finite-degree Z-closures of symmetric subsets of the Boolean cube in positive characteristic, revealing their algebraic structure and low computational complexity, with implications for combinatorics and circuit complexity.

Contribution

It provides a new characterization of finite-degree Z-closures for symmetric sets in positive characteristic, using symmetric polynomials with degree bounds.

Findings

Finite-degree Z-closures of symmetric sets are characterized explicitly.

These closures have low computational complexity.

Results unify and extend previous findings in combinatorics and circuit complexity.

Abstract

The finite-degree Zariski (Z-) closure is a classical algebraic object, that has found a key place in several applications of the polynomial method in combinatorics. In this work, we characterize the finite-degree Z-closures of a subclass of symmetric sets (subsets that are invariant under permutations of coordinates) of the Boolean cube, in positive characteristic. Our results subsume multiple statements on finite-degree Z-closures that have found applications in extremal combinatorial problems, for instance, pertaining to set systems (Heged\H{u}s, Stud. Sci. Math. Hung. 2010; Heged\H{u}s, arXiv 2021), and Boolean circuits (Hr\v{u}bes et al., ICALP 2019). Our characterization also establishes that for the subclasses of symmetric sets that we consider, the finite-degree Z-closures have low computational complexity. A key ingredient in our characterization is a new variant of finite-degree Z-closures, defined using vanishing conditions on only symmetric polynomials satisfying a degree bound.