Covering Symmetric Sets of the Boolean Cube by Affine Hyperplanes

Abstract

Alon and Füredi (European J. Combin., 1993) proved that any family of hyperplanes that covers every point of the Boolean cube $\{0,1\}^n$ except one must contain at least $n$ hyperplanes. We obtain two extensions of this result, in characteristic zero, for hyperplane covers of symmetric sets of the Boolean cube (subsets that are closed under permutations of coordinates), as well as for `polynomial covers' of `weight-determined' sets of `strictly unimodal uniform' (SU$^2$) grids. As a main tool for solving our problems, we give a combinatorial characterization of (finite-degree) Zariski (Z-) closures of symmetric sets of the Boolean cube -- the Z-closure of a symmetric set is symmetric. In fact, we obtain a characterization that concerns, more generally, weight-determined sets of SU$^2$ grids. However, in this generality, our characterization is not of the Z-closures -- unlike over the Boolean cube, the Z-closure of a weight-determined set need not be weight-determined. We introduce a new closure operator exclusively for weight-determined sets -- the `(finite-degree) Z*-closure' -- defined to be the maximal weight-determined set in the Z-closure. (This coincides with the Z-closure over the Boolean cube, for symmetric sets.) We obtain a combinatorial characterization of the finite-degree Z*-closures of weight-determined sets of an SU$^2$ grid. This characterization may also be of independent interest. Indeed, as further applications, we (i) give an alternate proof of a lemma by Alon et al. (IEEE Trans. Inform. Theory, 1988), and (ii) characterize the `certifying degrees' of weight-determined sets. Over the Boolean cube, our above characterization can also be derived using a result of Bernasconi and Egidi (Inf. Comput., 1999). However, our proof is independent of this result, works for all SU$^2$ grids, and could be regarded as being more combinatorial.