2017
DOI: 10.1016/j.jsc.2016.11.005
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Linear algebra for computing Gröbner bases of linear recursive multidimensional sequences

Abstract: The so-called Berlekamp-Massey-Sakata algorithm computes a Gröbner basis of a 0-dimensional ideal of relations satisfied by an input table. It extends the Berlekamp-Massey algorithm to n-dimensional tables, for n > 1. We investigate this problem and design several algorithms for computing such a Gröbner basis of an ideal of relations using linear algebra techniques. The first one performs a lot of table queries and is analogous to a change of variables on the ideal of relations. As each query to the table can … Show more

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Cited by 15 publications
(10 citation statements)
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“…Over A, trying to deduce Ann( ) from Ann(( ) ≥0 ), for random vectors , ∈ A ×1 , presents a problem when Ann( ) does not have the Gorenstein property [16,25]. When Ann( ) has the Gorenstein property, it has been showed that Ann( ) can be recovered, with high probability, by using a bidimensional sequence with random initial conditions, provided K has large characteristic [5]. When it does not have the property, Ann( ) is still recoverable with a similar approach, but using several sequences [29].…”
Section: Minimal Polynomials Of Sparse Matricesmentioning
confidence: 99%
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“…Over A, trying to deduce Ann( ) from Ann(( ) ≥0 ), for random vectors , ∈ A ×1 , presents a problem when Ann( ) does not have the Gorenstein property [16,25]. When Ann( ) has the Gorenstein property, it has been showed that Ann( ) can be recovered, with high probability, by using a bidimensional sequence with random initial conditions, provided K has large characteristic [5]. When it does not have the property, Ann( ) is still recoverable with a similar approach, but using several sequences [29].…”
Section: Minimal Polynomials Of Sparse Matricesmentioning
confidence: 99%
“…Sakata extended this algorithm first to dimension 2 [33] and then to the general case > 1 [34]; see also Norton and Fitzpatrick's extension to > 1 [13]. Recent work includes variants of Sakata's algorithm such as one which handles relations that are satisfied by several sequences simultaneously [35], approaches relating the problem to the kernel of a multi-Hankel matrix and exploiting either fast linear algebra [5] or a process similar to Gram-Schmidt orthogonalization [27], and an algorithm relying directly on multivariate polynomial arithmetic [6]. As for the representation of the output, all these algorithms compute a Gröbner basis or a border basis of the annihilator.…”
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confidence: 99%
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“…In [3,4], the authors present the Scalar-FGLM algorithm, extending the matrix version of the BM algorithm for multidimensional sequences. It consists in computing the relations of the sequence through the computation of a maximal full-rank matrix of a multi-Hankel matrix, a multivariate generalization of a Hankel matrix.…”
Section: Related Workmentioning
confidence: 99%