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Communication Dans Un Congrès Année : 2016

Guessing Linear Recurrence Relations of Sequence Tuples and P-recursive Sequences with Linear Algebra

Jérémy Berthomieu
Jean-Charles Faugère

Résumé

Given several n-dimensional sequences, we first present an algorithm for computing the Gröbner basis of their module of linear recurrence relations. A P-recursive sequence (ui)iNn satisfies linear recurrence relations with polynomial coefficients in i, as defined by Stanley in 1980. Calling directly the aforementioned algorithm on the tuple of sequences ((ijui)iNn)j for retrieving the relations yields redundant relations. Since the module of relations of a P-recursive sequence also has an extra structure of a 0-dimensional right ideal of an Ore algebra, we design a more efficient algorithm that takes advantage of this extra structure for computing the relations. Finally, we show how to incorporate Gröbner bases computations in an Ore algebra Kt1,,tn,x1,,xn, with commutators xkxxxk=tktttk=tkxxtk=0 for k and tkxkxktk=xk, into the algorithm designed for P-recursive sequences. This allows us to compute faster the Gr\"obner basis of the ideal spanned by the first relations, such as in \textsc{2D}/\textsc{3D}-space walks examples.
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hal-01314266 , version 1 (11-05-2016)

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Jérémy Berthomieu, Jean-Charles Faugère. Guessing Linear Recurrence Relations of Sequence Tuples and P-recursive Sequences with Linear Algebra. 41st International Symposium on Symbolic and Algebraic Computation, Jul 2016, Waterloo, ON, Canada. pp.95-102, ⟨10.1145/2930889.2930926⟩. ⟨hal-01314266⟩
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