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Polynomial-Division-Based Algorithms for Computing Linear Recurrence Relations
arXiv - CS - Symbolic Computation Pub Date : 2021-07-06 , DOI: arxiv-2107.02582
Jérémy BerthomieuPolSys, Jean-Charles FaugèrePolSys

Sparse polynomial interpolation, sparse linear system solving or modular rational reconstruction are fundamental problems in Computer Algebra. They come down to computing linear recurrence relations of a sequence with the Berlekamp-Massey algorithm. Likewise, sparse multivariate polynomial interpolation and multidimensional cyclic code decoding require guessing linear recurrence relations of a multivariate sequence.Several algorithms solve this problem. The so-called Berlekamp-Massey-Sakata algorithm (1988) uses polynomial additions and shifts by a monomial. The Scalar-FGLM algorithm (2015) relies on linear algebra operations on a multi-Hankel matrix, a multivariate generalization of a Hankel matrix. The Artinian Gorenstein border basis algorithm (2017) uses a Gram-Schmidt process.We propose a new algorithm for computing the Gr{\"o}bner basis of the ideal of relations of a sequence based solely on multivariate polynomial arithmetic. This algorithm allows us to both revisit the Berlekamp-Massey-Sakata algorithm through the use of polynomial divisions and to completely revise the Scalar-FGLM algorithm without linear algebra operations.A key observation in the design of this algorithm is to work on the mirror of the truncated generating series allowing us to use polynomial arithmetic modulo a monomial ideal. It appears to have some similarities with Pad{\'e} approximants of this mirror polynomial.As an addition from the paper published at the ISSAC conferance, we give an adaptive variant of this algorithm taking into account the shape of the final Gr{\"o}bner basis gradually as it is discovered. The main advantage of this algorithm is that its complexity in terms of operations and sequence queries only depends on the output Gr{\"o}bner basis.All these algorithms have been implemented in Maple and we report on our comparisons.

中文翻译:

用于计算线性递归关系的基于多项式除法的算法

稀疏多项式插值、稀疏线性系统求解或模有理重构是计算机代数中的基本问题。他们归结为使用 Berlekamp-Massey 算法计算序列的线性递推关系。同样,稀疏多元多项式插值和多维循环码解码需要猜测多元序列的线性递推关系。有几种算法解决了这个问题。所谓的 Berlekamp-Massey-Sakata 算法 (1988) 使用多项式加法和单项式移位。Scalar-FGLM 算法 (2015) 依赖于对多 Hankel 矩阵的线性代数运算,这是 Hankel 矩阵的多元泛化。Artinian Gorenstein 边界基算法 (2017) 使用 Gram-Schmidt 过程。我们提出了一种新算法来计算 Gr{\" o}仅基于多元多项式算术的序列关系理想的更基础。该算法使我们既可以通过使用多项式除法重新审视 Berlekamp-Massey-Sakata 算法,又可以在没有线性代数运算的情况下彻底修改 Scalar-FGLM 算法。截断的生成级数允许我们使用多项式算术模一个单项式理想。它似乎与该镜像多项式的 Pad{\'e} 近似有一些相似之处。作为 ISSAC 会议上发表的论文的补充,我们给出了该算法的自适应变体,同时考虑了最终 Gr{\ “随着它的发现,逐渐增加了基础。
更新日期:2021-07-07
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