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

We reduce the problem of computing the rank and a nullspace basis of a univariate polynomial matrix to polynomial matrix multiplication. For an input n×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˜(nω d) operations, with ω the exponent of matrix multiplication over K, then the algorithm uses O˜(MM(n, d)) operations in K. For m×n matrices of rank r and degree d, the cost expression is O˜(nmrω−2 d). 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. Categories and Subject Descriptors: I.1[S...

• Type: Conference
• Year: 2005
• Where: ISSAC
• Authors: Arne Storjohann, Gilles Villard
• Keywords: ISSAC 2005, Matrix, Matrix Multiplication, Polynomial Matrix