Solving systems of linear equations over polynomials

Abstract

We consider a system of linear equations of the form A( x) X( x) = b( x), where A( x), b( x) are given m × n and m × 1 matrices with entries from Q [ x], the ring of polynomials in the variable x over the rationals. We provide a polynomial-time algorithm to find the general solution of this system over Q [ x]. This is accomplished by devising a polynomial-time algorithm to find the triangular canonical form (Hermite form) of the matrix A( x) using unimodular column operations. As applications we are able to give polynomial-time algorithms for finding the diagonal (Smith canonical) form of a polynomial matrix, testing whether two given matrices of rational entries are similar and for finding the invariant factors of a matrix of rationals.