A determinantal version of the frobenius-könig theorem

Abstract

Let F be a field and let {d 1,…,dk } be a set of independent indeterminates over F. Let A(d 1,…,dk ) be an n × n matrix each of whose entries is an element of F or a sum of an element of F and one of the indeterminates in {d 1,…,dk }. We assume that no d 1 appears twice in A(d 1,…,dk ). We show that if det A(d 1,…,dk ) = 0 then A(d 1,…,dk ) must contain an r × s submatrix B, with entries in F, so that r + s = n + p and rank B ⩽ p − 1: for some positive integer p.