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) 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(log ns)2 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) field operations, where ω is the exponent of matrix multiplication.