On Lin–Bose problem Original Research Article

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 Fset membership, variantAl×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[x1,…,xn] is the polynomial ring in n variables x1,…,xn over any coefficient field K. Then there exist matrices Gset membership, variantAl×l and F1set membership, variantAl×m such that F=GF1 with detG=d and F1 is a ZLP matrix. We provide an elementary proof to this problem, and treat non-full rank case.