$L$-balancing families

Abstract

Let p be a prime and let f ∈ F p [x 1 , . . . , x 2p ] be a polynomial. Suppose that f (v F ) = 0 for each F ⊆ [2p], where |F | = p and that f (0) = 0. Then deg(f ) ≥ p.We prove here the following generalization of their result.Let p be a prime and q = p α > 1, α ≥ 1. Let n > 0 be a positive integer and q − 1 ≤ d ≤ n − q + 1 be an integer. Let F be a field of characteristic p. Suppose that f (v F ) = 0 for each F ⊆ [n], whereLet t = 2d be an even number and L ⊆ [d − 1] be a given subset. We say that F ⊆ 2 [t] is an L-balancing family if for each F ⊆ [t], whereWe give a general upper bound for the size of an L-balancing family.