The Lin-Bose Problem

Abstract

This brief studies prime factorization problems of a multivariate polynomial matrix. We mainly investigate the LinBose problem proposed by Lin and Bose in the multidimensional system theory context. A simple proof to the Lin-Bose problem is presented in a much easier way to understand. As a by-product, we obtain some properties on modules generated by all the rows of a matrix.Index Terms-Factor prime matrix, matrix factorization, multidimensional systems, n-D polynomial matrix, polynomial ring.

On Serre Reduction of Multidimensional Systems

“…First, we give the following several well-known results about the factorization of nD polynomial matrices [10,17,19,20]. [17,20]).…”

“…In general, this involves Serre reduction of multivariate (nD) polynomial matrices to some simple equivalent forms, especially their Smith normal forms. Serre reduction can help to investigate the structural of nD polynomial matrices, which has wide applications in image, complex networked system, and other areas (see [6][7][8][9][10][11]).…”

Factorizations for a class of multivariate polynomial matrices

“…Meanwhile, Lin and Bose (2001) presented the Lin-Bose's conjecture: a matrix admits a zero prime matrix factorization if its all maximal reduced minors generate the unit ideal. This conjecture was proved in (Liu et al, 2014;Pommaret, 2001;Wang and Feng, 2004), so the problem of zero prime matrix factorizations have been completely solved. Subsequently, Wang and Kwong (2005) put forward an algorithm based on module theory to solve the problem of minor prime matrix factorizations.…”

Minor Prime Factorization for n‐D Polynomial Matrices over Arbitrary Coefficient Field

“…For more general linear functional systems, e.g., partial differential systems or delay-differential systems, the resulting system matrices are multivariate. The problem of multivariate (n-D) polynomial matrix has attracted much attention over the past decades because of its diverse applications in n-D systems, network theory, Wiener-Hopf equations, controls, and signal processing ( [2][3][4][5][6][7][8]); its factorizations have attracted much attention over the past decades because of their wide applications ( [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25]). Lin-Bose conjecture is one of the important problems of n-D polynomial matrix factorizations, which has been proved in [15][16][17][18][19][20].…”