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Conference Papers Year : 2022

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

Vincent Neiger
Thi Xuan Vu
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Abstract

Consider a matrix $\mathbf{F} \in \mathbb{K}[x]^{m \times n}$ of univariatepolynomials over a field $\mathbb{K}$. We study the problem of computing thecolumn rank profile of $\mathbf{F}$. To this end we first give an algorithmwhich improves the minimal kernel basis algorithm of Zhou, Labahn, andStorjohann (Proceedings ISSAC 2012). We then provide a second algorithm whichcomputes the column rank profile of $\mathbf{F}$ with a rank-sensitivecomplexity of $O\tilde{~}(r^{\omega-2} n (m+D))$ operations in $\mathbb{K}$.Here, $D$ is the sum of row degrees of $\mathbf{F}$, $\omega$ is the exponent ofmatrix multiplication, and $O\tilde{~}(\cdot)$ hides logarithmic factors.
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Dates and versions

hal-03580860 , version 1 (18-02-2022)
hal-03580860 , version 2 (10-05-2022)

Identifiers

  • HAL Id : hal-03580860 , version 2

Cite

George Labahn, Vincent Neiger, Thi Xuan Vu, Wei Zhou. Rank-Sensitive Computation of the Rank Profile of a Polynomial Matrix. ISSAC 2022, Jul 2022, Villeneuve-d'Ascq, France. ⟨hal-03580860v2⟩
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