Lisa Nicklasson: The Determinantal Matroid

Abstract

Consider a generic \((m \times n)\)-matrix of rank r over an algebraically closed field, where only a subset of the entries is known. We say that the matrix is finitely completable if there are finitely many ways to fill in the unknown entries so that the matrix has the prescribed rank \(r\). The finitely completable patterns provide the base sets of a matroid. Alternatively, the matroid can be defined algebraically in terms of elimination subideals in the ideal of \((r + 1)\)-minors. In this talk we will explore this matroid, and try to get a hold of its independent sets using commutative algebra and matroid theory.