Asymptotically fast polynomial matrix algorithms for multivariable systems

Abstract : We present the asymptotically fastest known algorithms for some basic problems on univariate polynomial matrices: rank, nullspace, determinant, generic inverse, reduced form. 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 (up to logarithmic factors and the size of the output).
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Claude-Pierre Jeannerod, Gilles Villard. Asymptotically fast polynomial matrix algorithms for multivariable systems. [Research Report] LIP RR-2005-36, Laboratoire de l'informatique du parallélisme. 2005, 2+12p. ⟨hal-02102206⟩

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