A polynomial method approach to zero-sum subsets in F_p^2

Abstract

We prove that every subset of F_p^2 having a nonempty intersection with each of the 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_p^2)=p+OL(F_p)_ 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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