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Pré-Publication, Document De Travail Année : 2017

Computing canonical bases of modules of univariate relations

Thi Xuan Vu

Résumé

We study the computation of relations between elements of a finite-dimensional $\mathbb{K}[x]$-module. The latter is a quotient $\mathbb{K}[x]^n/\mathcal{M}$ specified by a basis $\mathbf{M}\in\mathbb{K}[x]^{n\times n}$ of $\mathcal{M}$. Then, on input $\mathbf{F} \in \mathbb{K}[x]^{m\times n}$, we seek canonical bases of the set of relations $\mathbf{p} \in \mathbb{K}[x]^{1\times m}$ such that $\mathbf{p} \mathbf{F} = 0 \bmod \mathbf{M}$. This generalizes the computation of approximant bases, where the basis $\mathbf{M}$ is a diagonal of powers of $x$. Focusing on a Hermite basis $\mathbf{M}$, our algorithm exploits the triangular shape to follow the divide-and-conquer approach used in fast approximant basis computation. Besides recent techniques for this approach, we rely on high-order lifting to perform fast modular products of the form $\mathbf{P}\mathbf{F} \bmod \mathbf{M}$. Our algorithm has a cost bound of $\mathcal{O}\tilde{~}(m^{\omega-1}D + n^{\omega} D/m)$ operations in $\mathbb{K}$, where $D = \deg(\det(\mathbf{M}))$ is the dimension of $\mathbb{K}[x]^n/\mathcal{M}$, $\mathcal{O}\tilde{~}(\cdot)$ indicates that logarithmic factors are omitted, and $\omega$ is the exponent of matrix multiplication. To the best of our knowledge, this had previously only been achieved for a diagonal matrix $\mathbf{M}$. As a particular case, our algorithm computes the shifted Popov form of $\mathbf{M}$ within the same cost bound, up to logarithmic factors, as the previously fastest known algorithm, which is randomized.
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Dates et versions

hal-01457979 , version 1 (06-02-2017)
hal-01457979 , version 2 (30-05-2017)

Identifiants

  • HAL Id : hal-01457979 , version 1

Citer

Vincent Neiger, Thi Xuan Vu. Computing canonical bases of modules of univariate relations. 2017. ⟨hal-01457979v1⟩
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