On partially matchable subspaces in a field extension

Abstract

We formulate and prove linear counterparts of results on partial matchings between finite subsets in abelian groups. The chosen setting is a field extension $K⊊L$, where we introduce a notion of partial matching between finite-dimensional $K$-subspaces $A,B\subseteq L$. Our main results include (1) a characterization of those pairs $(A,B)$ that are partially matchable up to a specified defect, (2) a decomposition theorem for pairs $(A,B)$ having positive deficiency, and (3) an existence criterion for pairs having a prescribed dimension and satisfying a deficiency bound. We use these results to recover and extend various parts of this area of matching theory, emphasizing the close analogy between the group-theoretic and linear perspectives. Our approach blends algebraic techniques with tools from matroidal transversal theory, and utilizes a linearized version of the $e$-transform from additive number theory.

Minor expository changes and clarifications added. Remark 3.12 has been expanded. Comments welcome!