基于多项式除法的线性递推关系计算算法
Polynomial-Division-Based Algorithms for Computing Linear Recurrence Relations
摘要：
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。它们归结为用Berlekamp-Massey算法计算序列的线性递归关系。同样，稀疏多元多项式插值和多维循环码译码也需要猜测多元序列的线性递归关系，有几种算法可以解决这个问题。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）依赖于多汉克尔矩阵上的线性代数运算，这是汉克尔矩阵的多变量推广。Artinian Gorenstein边界基算法（2017）使用了Gram-Schmidt过程。我们提出了一种新的算法，用于计算仅基于多元多项式算术的序列关系理想的Gr{<$}bner基。该算法既可以通过多项式除法来重新审视Berlekamp-Massey-Sakata算法，又可以完全修正Scalar-FGLM算法，而不需要线性代数运算。作为在ISSAC会议上发表的论文的补充，我们给出了这种算法的一个自适应变体，它考虑了最终Gr{<$}bner基的形状，随着它的逐渐发现。该算法的主要优点是它的复杂性方面的操作和序列查询只依赖于输出Gr{<$}bner basis.All这些算法已实现在Maple和我们的比较报告。
英文摘要：
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{ö}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{é} 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{ö}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{ö}bner basis.All these algorithms have been implemented in Maple and we report on our comparisons.
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