Prime factorization of N-D polynomial matrices

The prime factorization problem of multivariate polynomial matrices was posed by Youla and Gnavi (IEEE Trans. Circuits and Sys., vol. 26, pp. 105-111) in 1979, and still remains open although there are some partial results covering certain special situations. In 1999, Z. Lin (J. Multidimensional Sys. and Sig. Proc., vol. 10, pp. 379-393) made a conjecture regarding prime factorizability of polynomial matrices in terms of their reduced minors. In 2001, Z. Lin and N.K. Bose (Linear Algebra and its Appl., vol. 338, pp. 125-138) studied several conjectures including earlier conjecture by Lin, and showed that they are indeed all equivalent. In this paper, we prove Lin´s conjecture, and thereby prove the correctness of all the other conjectures posed by Lin and Bose.