Let $f_1,\ldots,f_m$ be elements in a quotient $R^n / N$ which has finite dimension as a $K$-vector space, where $R = K[X_1,\ldots,X_r]$ and $N$ is an $R$-submodule of $R^n$. We address the problem of computing a Gr\"obner basis of the module of syzygies of $(f_1,\ldots,f_m)$, that is, of vectors $(p_1,\ldots,p_m) \in R^m$ such that $p_1 f_1 + \cdots + p_m f_m = 0$. An iterative algorithm for this problem was given by Marinari, M\"oller, and Mora (1993) using a dual representation of $R^n / N$ as the kernel of a collection of linear functionals. Following this viewpoint, we design a divide-and-conquer algorithm, which can be interpreted as a generalization to several variables of Beckermann and Labahn's recursive approach for matrix Pad\'e and rational interpolation problems. To highlight the interest of this method, we focus on the specific case of bivariate Pad\'e approximation and show that it improves upon the best known complexity bounds.

中文翻译：

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一种计算有限维合子的Gr\"obner基的分治算法

设 $f_1,\ldots,f_m$ 是商 $R^n / N$ 中的元素，商 $R^n / N$ 具有有限维数作为 $K$-向量空间，其中 $R = K[X_1,\ldots,X_r]$ 和 $ N$ 是 $R^n$ 的 $R$-子模块。我们解决了计算 $(f_1,\ldots,f_m)$ syzygies 模块的 Gr\"obner 基的问题，即向量 $(p_1,\ldots,p_m) \in R^m$ 这样的$p_1 f_1 + \cdots + p_m f_m = 0$。Marinari、M\"oller 和 Mora (1993) 使用 $R^n / N$ 的双重表示作为内核给出了该问题的迭代算法线性泛函的集合。根据这个观点，我们设计了一个分而治之的算法，它可以解释为贝克曼和拉巴恩对矩阵 Pad\'e 和有理插值问题的递归方法的几个变量的推广。为了突出这种方法的兴趣，