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 be expensive, we design a second algorithm requiring fewer queries, in general. This FGLM-like algorithm allows us to compute the relations of the table by extracting a full rank submatrix of a multi-Hankel matrix (a multivariate generalization of Hankel matrices). Under some additional assumptions, we make a third, adaptive, algorithm and reduce further the number of table queries. Then, we relate the number of queries of this third algorithm to the geometry of the final staircase and we show that it is essentially linear in the size of the output when the staircase is convex. As a direct application to this, we decode n-cyclic codes, a generalization in dimension n of Reed Solomon codes. We show that the multi-Hankel matrices are heavily structured when using the LEX ordering and that we can speed up the computations using fast algorithms for quasi-Hankel matrices. Finally, we design algorithms for computing the generating series of a linear recursive table.

Algorithms for Linearly Recurrent Sequences of Truncated Polynomials

“…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].…”

“…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.…”

“…3]. Apart from this, to the best of our knowledge previous work has = 1 and considers -dimensional sequences over K for an arbitrary ≥ 2 [5,6,27]. Complexity in this -variate context is often expressed using the degree of the considered zero-dimensional ideal; here, ≤ ≤ and a minimal Gröbner basis or a border basis will have at most min( , ) + 1 elements.…”

“…Complexity in this -variate context is often expressed using the degree of the considered zero-dimensional ideal; here, ≤ ≤ and a minimal Gröbner basis or a border basis will have at most min( , ) + 1 elements. The Scalar-FGLM algorithm has cost˜( opt ) [5,Prop. 16].…”

A Polynomial-Division-Based Algorithm for Computing Linear Recurrence Relations

“…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.…”