A Generalization of Serre's Conjecture and Some Related Issues

Several topics concerned with multivariate polynomial matrices like unimodular matrix completion, matrix determinantal or primitive factorization, matrix greatest common factor existence and subsequent extraction along with relevant primeness and coprimeness issues are related to a conjecture which may be viewed as a type of generalization of the original Serre problem (conjecture) solved nonconstructively in 1976 and constructively, more recently. This generalized Serre conjecture is proved to be equivalent to several other unsettled conjectures and, therfore, all these conjectures constitute a complete set in the sense that solution to any one also solves all the remaining