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.
翻译:粗微的单线性内插、 稀疏的线性系统解析或模块化的合理重建是计算机代数中的根本问题。 它们可以用来计算与 Berlekamp- Masssey 算法序列的线性重复关系。 同样, 稀疏的多变多式多式多元性内插和多维循环代码解码要求猜测多变序列的线性重复关系。 量算法可以解决这个问题。 所谓的 Berlekamp- Masssey- Sakata 算法( 1988) 使用多式的多元体内添加和变换。 Scalar- FGLM 算法( 2015) 依靠多动体内的线性变数性变数, 汉克尔矩阵矩阵矩阵的多变数性变法化。 Artinian Gorestein 边基算法( 2017) 使用Gram- Schmidt 进程。 我们提议一个新的算法基础, 仅以多变数性多式的多式内数变数性算法变算算法为基础, 这个算法可以让我们重新审视Berlalamp- massyalalalalal- 的算算算算算算算算法。 这个算法的算算法, 这个算算算算算法, 这个算算法, 这个算法的算法的算法的算法, 的算法, 这个算法的算法, 它的算算法, 的算法, 开始一个算法的算法的算法的算法的算法的算算法的算算法, 的算法, 的算算法的算法的算法的算法的算法的算法的算法的算法的算法的算法的算法的算法的算法的算法, 的算法的算法的算法的算法的算法的算法的算法的算法的算法的算法的算法的算法的算法的算法的算法的算法的算法的算法的算法的算法的算法的算法的算法的算法的算法的算法的算法的算法的算法的算法的算法的算法的