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Alon and F\"uredi (European J. Combin. 1993) gave a tight bound for thefollowing hyperplane covering problem: find the minimum number of hyperplanesrequired to cover all points of the n-dimensional hypercube {0,1}^n except theorigin. Their proof is among the early instances of the polynomial method,which considers a natural polynomial (a product of linear factors) associatedto the hyperplane arrangement, and gives a lower bound on its degree, whilstbeing oblivious to the (product) structure of the polynomial. Thus, their proofgives a lower bound for a weaker polynomial covering problem, and it turns outthat this bound is tight for the stronger hyperplane covering problem.  In a similar vein, solutions to some other hyperplane covering problems wereobtained, via solutions of corresponding weaker polynomial covering problems,in some special cases in the works of the fourth author (Electron. J. Combin.2022), and the first three authors (Discrete Math. 2023). In this work, webuild on these and solve a hyperplane covering problem for general symmetricsets of the hypercube, where we consider hyperplane covers with highermultiplicities. We see that even in this generality, it is enough to solve thecorresponding polynomial covering problem. Further, this seems to be the limitof this approach as far as covering symmetry preserving subsets of thehypercube is concerned. We gather evidence for this by considering the class ofblockwise symmetric sets of the hypercube (which is a strictly larger classthan symmetric sets), and note that the same proof technique seems to onlysolve the polynomial covering problem.

On higher multiplicity hyperplane and polynomial covers for symmetry preserving subsets of the hypercube