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 Gröbner 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 Padé approximants of this mirror polynomial.

As an addition from the paper published at the ISSAC conference, we give an adaptive variant of this algorithm taking into account the shape of the final Gröbner 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 Gröbner basis.

All these algorithms have been implemented in Maple and we report on our comparisons.

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。的标量FGLM算法（2015）依赖于线性代数运算上的多汉克尔矩阵，汉克尔矩阵的多元推广。Artinian Gorenstein 边界基算法 (2017) 使用 Gram-Schmidt 过程。

我们提出了一种新算法，用于计算仅基于多元多项式算术的序列关系理想的 Gröbner 基。该算法允许我们通过使用多项式除法重新审视 Berlekamp-Massey-Sakata 算法，并在没有线性代数运算的情况下完全修改Scalar-FGLM算法。

该算法设计中的一个关键观察是在截断生成级数的镜像上工作，允许我们使用多项式算术对单项式理想取模。它似乎与这个镜像多项式的 Padé 近似有一些相似之处。

作为在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，并在发现时逐渐考虑到最终 Gröbner 基的形状。该算法的主要优点是其在操作和序列查询方面的复杂性仅取决于输出 Gröbner 基。

所有这些算法都在Maple 中实现，我们报告了我们的比较。