Asymptotically fast polynomial matrix algorithms for multivariable systems

Abstract

We present the asymtotically fastest known algorithms for some basic problems on univariate polynomial matrices: rank; nullspace; determinant; generic inverse reduced form (Giorgi et al. 2003, Storjohann 2003, Jeannerod and Villard 2005, Storjohann and Villard 2005). We show that they essentially can be reduced to two computer algebra techniques, minimal basis computations and matrix fraction expansion/reconstruction, and to polynomial matrix multiplication. Such reductions eventually imply that all these problems can be solved in about the same amount of time as polynomial matrix multiplication. The algorithms are deterministic, or randomized with certified output in a Las Vegas fashion.