Essentially optimal computation of the inverse of generic polynomial matrices

We present an inversion algorithm for nonsingular n×n matrices whose entries are degree d polynomials over a field. The algorithm is deterministic and, when n is a power of two, requires O∼(n3d) field operations for a generic input; the soft-O notation O∼ indicates some missing log(nd) factors. Up to such logarithmic factors, this asymptotic complexity is of the same order as the number of distinct field elements necessary to represent the inverse matrix.