The odd case of Rota's bases conjecture

Abstract

The paper links four conjectures:(1)(Rota's bases conjecture): For any system A=(A1,…,An) of non-singular real valued matrices the multiset of all columns of matrices in A can be decomposed into n independent systems of representatives of A.(2)(Alon–Tarsi): For even n, the number of even n×n Latin squares differs from the number of odd n×n Latin squares.(3)(Stones–Wanless, Kotlar): For all n, the number of even n×n Latin squares with the identity permutation as first row and first column differs from the number of odd n×n Latin squares of this type.(4)(Aharoni–Berger): Let M and N be two matroids on the same vertex set, and let A1,…,An be sets of size n+1 belonging to M∩N. Then there exists a set belonging to M∩N meeting all Ai. Huang and Rota [8] and independently Onn [11] proved that for any n (2) implies (1). We prove equivalence between (2) and (3). Using this, and a special case of (4), we prove the Huang–Rota–Onn theorem for n odd and a restricted class of input matrices: assuming the Alon–Tarsi conjecture for n−1, Rota's conjecture is true for any system of non-singular real valued matrices where one of them is non-negative and the remaining have non-negative inverses.