On Lin–Bose problem

Abstract

This paper study generalized Serre problem proposed by Lin and Bose in multidimensional system theory context [Multidimens. Systems and Signal Process. 10 (1999) 379; Linear Algebra Appl. 338 (2001) 125]. This problem is stated as follows. Let F∈ A l× m be a full row rank matrix, and d be the greatest common divisor of all the l× l minors of F. Assume that the reduced minors of F generate the unit ideal, where A= K[ x 1,…, x n ] is the polynomial ring in n variables x 1,…, x n over any coefficient field K. Then there exist matrices G∈ A l× l and F 1∈ A l× m such that F= GF 1 with det G= d and F 1 is a ZLP matrix. We provide an elementary proof to this problem, and treat non-full rank case.