"Covering almost all the layers of the hypercube with multiplicities" by Arijit Ghosh, Chandrima Kayal et al.
 

Covering almost all the layers of the hypercube with multiplicities

Article Type

Research Article

Publication Title

Discrete Mathematics

Abstract

Given a hypercube Qn:={0,1}n in Rn and k∈{0,…,n}, the k-th layer Qkn of Qn denotes the set of all points in Qn whose coordinates contain exactly k many ones. For a fixed t∈N and k∈{0,…,n}, let P∈R[x1,…,xn] be a polynomial that has zeroes of multiplicity at least t at all points of Qn∖Qkn, and P has zeros of multiplicity exactly t−1 at all points of Qkn. 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 H1,…,Hm in Rn, where m=max⁡{k,n−k}+2t−2, such that every point of Qkn will be covered exactly t−1 times, and every other point of Qn 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 (1993) [1]. Using the above family of hyperplanes we disprove a conjecture of Venkitesh (2022) [23] on exactly covering symmetric subsets of hypercube Qn 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.

DOI

https://10.1016/j.disc.2023.113397

Publication Date

7-1-2023

Comments

Open Access, Green

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