有限维合偶格罗布纳基的分治算法
A divide-and-conquer algorithm for computing Gröbner bases of syzygies in finite dimension
摘要：
设$f_1,\ldots,f_m$是作为$K$ -向量空间的有限维商$R^n / N$中的元素，其中$R = K[X_1,\ldots,X_r]$和$N$是$R^n$的$R$ -子模.我们解决的问题计算的Gröbner基础的模块的syzygies的$(f_1,\ldots,f_m)$，即向量$(p_1,\ldots,p_m) \in R^m$，使$p_1 f_1 + \cdots + p_m f_m = 0$。Marinari，Möller，and Mora（1993）给出了这个问题的迭代算法，使用$R^n / N$的对偶表示作为线性泛函集合的核。根据这一观点，我们设计了一个分治算法，它可以被解释为一个推广到多个变量的Beckermann和Labahn的递归方法矩阵Padé和有理插值问题。为了突出这种方法的兴趣，我们专注于二元Padé近似的特定情况下，并表明它提高了最知名的复杂性界限。
英文摘要：
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 Gröbner 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, Möller, 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 Padé and rational interpolation problems. To highlight the interest of this method, we focus on the specific case of bivariate Padé approximation and show that it improves upon the best known complexity bounds.
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