Computing canonical bases of modules of univariate relations
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.
Origine | Fichiers produits par l'(les) auteur(s) |
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