Abstract
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.
Highlights
The Berlekamp–Massey algorithm (BM), introduced by Berlekamp in 1968 [2] and Massey in 1969 [24] is a fundamental algorithm in Coding Theory, [9, 20], and Computer Algebra
In 1988, Sakata extended the BM algorithm to dimension n. This algorithm, known as the Berlekamp–Massey–Sakata algorithm (BMS), can be used to compute a Grobner basis of the zero-dimensional ideal of the relations satisfied by a sequence, [27, 28, 29]
The latest versions of the Sparse-FGLM algorithm rely heavily on the efficiency of the BMS algorithm to compute the change of ordering of a Grobner basis, [16, 17]
Summary
The Berlekamp–Massey algorithm (BM), introduced by Berlekamp in 1968 [2] and Massey in 1969 [24] is a fundamental algorithm in Coding Theory, [9, 20], and Computer Algebra. It allows one to perform efficiently sparse polynomial interpolation, sparse linear system solving or modular rational reconstruction. In 1988, Sakata extended the BM algorithm to dimension n. This algorithm, known as the Berlekamp–Massey–Sakata algorithm (BMS), can be used to compute a Grobner basis of the zero-dimensional ideal of the relations satisfied by a sequence, [27, 28, 29]. To dimension 1, the BMS algorithm allows one to decode cyclic codes in dimension n > 1, an extension of Reed–Solomon’s codes. The latest versions of the Sparse-FGLM algorithm rely heavily on the efficiency of the BMS algorithm to compute the change of ordering of a Grobner basis, [16, 17]
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