Linear Algebra for Computing Gröbner Bases of Linear Recursive Multidimensional Sequences

Sakata generalized the Berlekamp--Massey algorithm to n dimensions in~1988. The Berlekamp--Massey--Sakata (BMS) algorithm can be used for finding a Grbner basis of a 0-dimensional ideal of relations verified by a table. We investigate this problem usingö linear algebra techniques, with motivations such as accelerating change of basis algorithms (FGLM) or improving their complexity. We first define and characterize multidimensional linear recursive sequences for 0-dimensional ideals. Under genericity assumptions, we propose a randomized preprocessing of the table that corresponds to performing a linear change of coordinates on the polynomials associated with the linear recurrences. This technique then essentially reduces our problem to using the efficient 1-dimensional Berlekamp--Massey (BM) algorithm. However, the number of probes to the table in this scheme may be elevated. We thus consider the table in the black-box model: we assume probing the table is expensive and we minimize the number of probes to the table in our complexity model. We produce an FGLM-like algorithm for finding the relations in the table, which lets us use linear algebra techniques. Under some additional assumptions, we make this algorithm adaptive and reduce further the number of table probes. This number can be estimated by counting the number of distinct elements in a multi-Hankel matrix (a multivariate generalization of Hankel matrices); we can relate this quantity with the geometry of the final staircase. Hence, in favorable cases such as convex ones, the complexity is essentially linear in the size of the output. Finally, when using the LEX ordering, we can make use of fast structured linear algebra similarly to the Hankel interpretation of Berlekamp--Massey.