A weak version of Rota's basis conjecture for odd dimensions

TL;DR

This paper extends the Alon-Tarsi Latin square conjecture to odd dimensions and demonstrates that its validity implies a weaker form of Rota's basis conjecture, asserting the existence of multiple disjoint independent transversals.

Contribution

It introduces a new extension of the Latin square conjecture for odd dimensions and links it to a weaker form of Rota's basis conjecture using a modified determinantal identity.

Findings

Extension of Alon-Tarsi conjecture to odd dimensions

Implication of conjecture validity on Rota's basis conjecture

Existence of n-1 disjoint independent transversals in certain bases

Abstract

The Alon-Tarsi Latin square conjecture is extended to odd dimensions by stating it for reduced Latin squares (Latin squares having the identity permutation as their first row and first column). A modified version of Onn's colorful determinantal identity is used to show how the validity of this conjecture implies a weak version of Rota's basis conjecture for odd dimensions, namely that a set of $n$ bases in $\mathbb{R}^n$ has $n-1$ disjoint independent transversals.