Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Covering Symmetric Sets of the Boolean Cube by Affine Hyperplanes   Alon and F\"uredi (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.

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The categories of this paper are: solv-int
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The categories of this paper are: eess
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The categories of this paper are: comp-gas
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The categories of this paper are: chao-dyn
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The categories of this paper are: bayes-an
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The categories of this paper are: math-ph
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The categories of this paper are: econ
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The categories of this paper are: cmp-lg
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The categories of this paper are: hep-ex
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The categories of this paper are: plasm-ph
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The categories of this paper are: nucl-th
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The categories of this paper are: q-fin
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The categories of this paper are: dg-ga
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The categories of this paper are: q-alg
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The categories of this paper are: atom-ph
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The categories of this paper are: nlin