Linear Algebra for Computing Gröbner Bases of Linear Recursive Multidimensional Sequences

International audienceSakata generalized the Berlekamp -- Massey algorithm to $n$ dimensions in~1988. The Berlekamp -- Massey -- Sakata (BMS)algorithm can be used for finding a Gröbner basis of a $0$-dimensionalideal of relations verified by a table. We investigate this problem usinglinear algebra techniques, with motivations such as accelerating change ofbasis algorithms (FGLM) or improving their complexity.We first define and characterize multidimensional linear recursive sequencesfor $0$-dimensional ideals.Under genericity assumptions, we propose a randomized preprocessing of thetable that corresponds to performing a linear change of coordinates on thepolynomials associated with the linear recurrences. This technique thenessentially 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 \emph{black-box} model: we assume probing thetable is expensive and we minimize the number of probes to the table in ourcomplexity model.We produce an FGLM-like algorithm for finding the relations in thetable, which lets us use linear algebra techniques. Under some additionalassumptions, we make this algorithm adaptive and reduce further the numberof table probes.This number can be estimated by counting the number of distinct elements in amulti-Hankel matrix (a multivariate generalization of Hankel matrices); we canrelate this quantity with the \emph{geometry} of the final staircase. Hence,in favorable cases such as convex ones, the complexity is essentially linear inthe size of the output. Finally, when using the \textsc{lex} ordering, we canmake use of fast structured linear algebra similarly to the Hankelinterpretation of Berlekamp -- Massey

Linear Algebra for Computing Gröbner Bases of Linear Recursive Multidimensional Sequences

Special issue on the conference ISSAC 2015: Symbolic computation and computer algebraInternational audienceThe so-called Berlekamp~-- Massey~-- Sakata algorithmcomputes a Gröbner basis of a $0$-dimensional ideal of relations satisfied by an inputtable. It extends the Berlekamp~-- Massey algorithmto $n$-dimensional tables, for $n>1$.We investigate this problem and design several algorithms forcomputing such a Gröbner basis of an ideal of relations using linearalgebra techniques.The first one performs a lot of table queries andis analogous to a change of variables on the ideal of relations.As each query to the table can be expensive,we design a second algorithmrequiring fewer queries, in general.This \textsc{FGLM}-like algorithm allows us to compute the relations of thetable by extracting a full rank submatrix of a \emph{multi-Hankel}matrix (a multivariate generalization of Hankel matrices).Under someadditional assumptions, we make a third, adaptive, algorithm and reducefurther the number of table queries.Then, we relate the number of queries ofthis third algorithm to the\emph{geometry} of the final staircase and we show that it isessentially linear in the size of the output when the staircase is convex.As a direct application to this, we decode $n$-cyclic codes, ageneralization in dimension $n$ of Reed Solomon codes. We show that the multi-Hankelmatrices are heavily structured when using the \textsc{LEX} orderingand that we can speed up the computations using fast algorithms forquasi-Hankel matrices.Finally, we designalgorithms for computing the generating series of a linear recursivetable