On polynomials that vanish to high order on most of the hypercube

Abstract: Motivated by higher vanishing multiplicity generalizations of Alon's Combinatorial Nullstellensatz and its applications, we study the following problem: for fixed k and n large with respect to k, what is the minimum possible degree of a polynomial P in R[x_1,...,x_n] such that P(0,…,0) is non-zero and such that P has zeroes of multiplicity at least k at all points in ${0,1}^n$ except the origin? For k=1, a classical theorem of Alon and Füredi states that the minimum possible degree of such a polynomial equals n. We solve the problem for all k>1, proving that the answer is n+2k−3. Joint work with Yuval Wigderson.

combinatoricsprobability

Audience: researchers in the topic

Extremal and probabilistic combinatorics webinar

Series comments: We've added a password: concatenate the 6 first prime numbers (hence obtaining an 8-digit password).