# A Polynomial Method Approach to Zero-Sum Subsets in $\mathbb{F}_{p}^{2}$

## TL;DR

This paper proves that subsets of _p^2 meeting all lines through the origin contain zero-sum subsets, using a polynomial method inspired by the Combinatorial Nullstellensatz, advancing understanding of Olson constants.

## Contribution

The paper introduces a polynomial method approach to establish the existence of zero-sum subsets in _p^2 under specific geometric conditions, extending Olson constant results.

## Key findings

- Every subset meeting all lines through the origin has a zero-sum subset.
- The proof employs a polynomial method inspired by the Combinatorial Nullstellensatz.
- Extends Olson constant results for _p^2.

## Abstract

In this paper we prove that every subset of $\mathbb{F}_p^2$ 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 $OL(\mathbb{F}_{p}^{2})=p+OL(\mathbb{F}_{p})-1$, 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.