The Alon-Jaeger-Tarsi conjecture via group ring identities

TL;DR

This paper proves the Alon-Jaeger-Tarsi conjecture for large finite fields, demonstrating the existence of a vector avoiding zero components after certain linear transformations, using group ring identities.

Contribution

It introduces a novel approach using group ring identities to resolve the conjecture for sufficiently large primes.

Findings

Confirmed the conjecture for fields with size greater than 61 and not equal to 79.

Established the existence of vectors avoiding zero components after linear transformation.

Extended the validity of the conjecture to a broad class of finite fields.

Abstract

In this paper we resolve the Alon-Jaeger-Tarsi conjecture for sufficiently large primes. Namely, we show that for any finite field $\mathbb{F}$ of size $61<|\mathbb F|\ne 79$ and any nonsingular matrix $M$ over $\mathbb{F}$ there exists a vector $x$ such that neither $x$ nor $Ax$ has a 0 component.