  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.
