General equivalence problem for multivariate polynomial matrices

Establish general criteria characterizing when an l×m polynomial matrix with entries in the multivariate polynomial ring K[x1, x2, ..., xn] for n ≥ 2 is equivalent over K[x1, x2, ..., xn] to its Smith normal form via unimodular row and column operations, thereby resolving the equivalence problem in the multivariate (non-PID) setting.

Background

The paper contrasts the well-understood univariate case—where K[x] is a PID and every polynomial matrix is equivalent to its Smith normal form—with the multivariate case, where K[x1, ..., xn] is not a PID. This non-PID structure obstructs a general theory guaranteeing equivalence to Smith normal forms and leaves broad aspects of the equivalence problem unresolved.

The authors summarize two major classes of multivariate polynomial matrices for which the equivalence problem has been completely solved. They then emphasize that beyond these classes, the equivalence problem remains open for many other multivariate matrices, highlighting a substantial gap in the general theory.