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arXiv · 2111.05445

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

Srikanth Srinivasan·S. Venkitesh
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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.

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Srikanth Srinivasan, S. Venkitesh. 2021-11-09. On Vanishing Properties of Polynomials on Symmetric Sets of the Boolean Cube, in Positive Characteristic. https://arxiv.org/abs/2111.05445

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