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

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