A Polynomial Method Approach to Zero-Sum Subsets in F^2_p

Abstract

In this paper we prove that every subset of F^2_p meeting all p+1 lines passing through the origin has a zero-sum subset. This is motivated by a result of Gao, Ruzsa and Thangadurai which states that OL(F^2_p) = p+OL(F_p)−1, for sufficiently large primes p. Here OL(G) denotes the so-called Olson constant of the additive group G and represents the smallest integer such that no subset of cardinality OL(G) is zero-sum-free. Our proof is in the spirit of the Combinatorial Nullstellensatz.

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