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The finite-degree Zariski (Z-) closure is a classical algebraic object, thathas found a key place in several applications of the polynomial method incombinatorics. In this work, we characterize the finite-degree Z-closures of asubclass of symmetric sets (subsets that are invariant under permutations ofcoordinates) of the Boolean cube, in positive characteristic.  Our results subsume multiple statements on finite-degree Z-closures that havefound applications in extremal combinatorial problems, for instance, pertainingto set systems (Heged\H{u}s, Stud. Sci. Math. Hung. 2010; Heged\H{u}s, arXiv2021), and Boolean circuits (Hr\v{u}bes et al., ICALP 2019). Ourcharacterization also establishes that for the subclasses of symmetric setsthat we consider, the finite-degree Z-closures have low computationalcomplexity.  A key ingredient in our characterization is a new variant of finite-degreeZ-closures, defined using vanishing conditions on only symmetric polynomialssatisfying a degree bound.

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