Sharp proximity bounds for group-labelled matroid bases

Let M be a finite matroid whose elements are labelled by several abelian groups, with a finite set of forbidden sums in each coordinate. We prove that if M is representable over a field and has a feasible basis, then every basis of M is within ∏ₜ(|Fₜ|+1)-1 exchanges of a feasible basis. The bound is sharp for every choice of the forbidden-set sizes, over every representation field.The proof uses exterior contraction over a commutative ring and interpolation in independent copies of group algebras. We also obtain the same bound for SIBO matroids and for the class generated from complex-representable and SIBO matroids by direct sums and two-sums. For arbitrary matroids, the bound holds when the joint labels on the starting basis are constant outside a common set of at most two elements.