Computing the rank and a small nullspace basis of a polynomial matrix

  • Authors:

  • Arne Storjohann;Gilles Villard

  • Affiliations:

  • University of Waterloo, Waterloo, ON, Canada;CNRS, LIP, École Normale Supérieure de Lyon, Lyon Cedex, France

  • Venue:
  • Proceedings of the 2005 international symposium on Symbolic and algebraic computation
  • Year:
  • 2005

Quantified Score

Hi-index 0.00

Visualization

Abstract

We reduce the problem of computing the rank and a null-space basis of a univariate polynomial matrix to polynomial matrix multiplication. For an input n x n matrix of degree, d over a field K we give a rank and nullspace algorithm using about the same number of operations as for multiplying two matrices of dimension, n and degree, d. If the latter multiplication is done in MM(n,d)= O~(nωd operations, with ω the exponent of matrix multiplication over K, then the algorithm uses O~MM(n,d) operations in, K. For m x n matrices of rank r and degree d, the cost expression is O(nmr ω-2d). The soft-O notation O~ indicates some missing logarithmic factors. The method is randomized with Las Vegas certification. We achieve our results in part through a combination of matrix Hensel high-order lifting and matrix minimal fraction reconstruction, and through the computation of minimal or small degree vectors in the nullspace seen as a K[x]-module.