Straight-line computation of the polynomial matrix inverse

We present an inversion algorithm for nonsingular n × n matrices whose entries are degree d polynomials over a field. The algorithm is deterministic and requires O (̃nd) field operations for a generic input. The “soft Oh” notation O ̃indicates some missing log(nd) factors. The polynomial matrix inverse can thus be computed by a straight-line program whose size is — asymptotically and up to logarithmic factors — the size of its output.