  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.
