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# Computing syzygies in finite dimension using fast linear algebra

Abstract : We consider the computation of syzygies of multivariate polynomials in a finite-dimensional setting: for a $\mathbb{K}[X_1,\dots,X_r]$-module $\mathcal{M}$ of finite dimension $D$ as a $\mathbb{K}$-vector space, and given elements $f_1,\dots,f_m$ in $\mathcal{M}$, the problem is to compute syzygies between the $f_i$'s, that is, polynomials $(p_1,\dots,p_m)$ in $\mathbb{K}[X_1,\dots,X_r]^m$ such that $p_1 f_1 + \dots + p_m f_m = 0$ in $\mathcal{M}$. Assuming that the multiplication matrices of the $r$ variables with respect to some basis of $\mathcal{M}$ are known, we give an algorithm which computes the reduced Gr\"obner basis of the module of these syzygies, for any monomial order, using $O(m D^{\omega-1} + r D^\omega \log(D))$ operations in the base field $\mathbb{K}$, where $\omega$ is the exponent of matrix multiplication. Furthermore, assuming that $\mathcal{M}$ is itself given as $\mathcal{M} = \mathbb{K}[X_1,\dots,X_r]^n/\mathcal{N}$, under some assumptions on $\mathcal{N}$ we show that these multiplication matrices can be computed from a Gr\"obner basis of $\mathcal{N}$ within the same complexity bound. In particular, taking $n=1$, $m=1$ and $f_1=1$ in $\mathcal{M}$, this yields a change of monomial order algorithm along the lines of the FGLM algorithm with a complexity bound which is sub-cubic in $D$.
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https://hal-unilim.archives-ouvertes.fr/hal-02392488
Contributeur : Vincent Neiger <>
Soumis le : vendredi 19 juin 2020 - 12:00:26
Dernière modification le : vendredi 26 juin 2020 - 03:30:14

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### Citation

Vincent Neiger, Éric Schost. Computing syzygies in finite dimension using fast linear algebra. Journal of Complexity, Elsevier, In press, ⟨10.1016/j.jco.2020.101502⟩. ⟨hal-02392488v2⟩

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