Almost covering all the layers of hypercube with multiplicities (original) (raw)

2022, arXiv (Cornell University)

Given a hypercube Q n := {0, 1} n in R n and k ∈ {0,. .. , n}, the k-th layer Q n k of Q n denotes the set of all points in Q n whose coordinates contain exactly k many ones. For a fixed t ∈ N and k ∈ {0,. .. , n}, let P ∈ R[x 1 ,. .. , x n ] be a polynomial that has zeroes of multiplicity at least t at all points of Q n \ Q n k , and P has zeros of multiplicity exactly t − 1 at all points of Q n k. In this short note, we show that deg(P) ≥ max {k, n − k} + 2t − 2. Matching the above lower bound we give an explicit construction of a family of hyperplanes H 1 ,. .. , H m in R n , where m = max {k, n − k} + 2t − 2, such that every point of Q n k will be covered exactly t − 1 times, and every other point of Q n will be covered at least t times. Note that putting k = 0 and t = 1, we recover the much celebrated covering result of Alon and Füredi (European Journal of Combinatorics, 1993). Using the above family of hyperplanes we disprove a conjecture of Venkitesh (The Electronic Journal of Combinatorics, 2022) on exactly covering symmetric subsets of hypercube Q n with hyperplanes. To prove the above results we have introduced a new measure of complexity of a subset of the hypercube called index complexity which we believe will be of independent interest. We also study a new interesting variant of the restricted sumset problem motivated by the ideas behind the proof of the above result.