Rank-Sensitive Computation of the Rank Profile of a Polynomial Matrix

02/18/2022
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by   George Labahn, et al.
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Consider a matrix ๐…โˆˆ๐•‚[x]^m ร— n of univariate polynomials over a field ๐•‚. We study the problem of computing the column rank profile of ๐…. To this end we first give an algorithm which improves the minimal kernel basis algorithm of Zhou, Labahn, and Storjohann (Proceedings ISSAC 2012). We then provide a second algorithm which computes the column rank profile of ๐… with a rank-sensitive complexity of O(r^ฯ‰-2 n (m+D)) operations in ๐•‚. Here, D is the sum of row degrees of ๐…, ฯ‰ is the exponent of matrix multiplication, and O(ยท) hides logarithmic factors.

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