A deterministic algorithm for inverting a polynomial matrix

Improved cost estimates are given for the problem of computing the inverse of an n×n matrix of univariate polynomials over a field. A deterministic algorithm is demonstrated that has worst case complexity (n3s)1+o(1) field operations, where s≥1 is an upper bound for the average column degree of the input matrix. Here, the “+o(1)” in the exponent indicates a missing factor c1(logns)c2 for positive real constants c1 and c2. As an application we show how to compute the largest invariant factor of the input matrix in (nωs)1+o(1) field operations, where ω is the exponent of matrix multiplication.