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

Jérémy Berthomieu 1 Jean-Charles Faugère 1 
1 PolSys - Polynomial Systems
LIP6 - Laboratoire d'Informatique de Paris 6, Inria de Paris
Abstract : 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 $(u_{\mathbf{i}})_{\mathbf{i}\in\mathbb{N}^n}$ satisfies linear recurrence relations with polynomial coefficients in $\mathbf{i}$, as defined by Stanley in 1980. Calling directly the aforementioned algorithm on the tuple of sequences $\left((\mathbf{i}^{\mathbf{j}}\,u_{\mathbf{i}})_{\mathbf{i}\in\mathbb{N}^n}\right)_{\mathbf{j}}$ 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 $\mathbb{K}\langle t_1,\ldots,t_n,x_1,\ldots,x_n\rangle$, with commutators $x_k\,x_{\ell}-x_{\ell}\,x_k=t_k\,t_{\ell}-t_{\ell}\,t_k=t_k\,x_{\ell}-x_{\ell}\,t_k=0$ for $k\neq\ell$ and $t_k\,x_k-x_k\,t_k=x_k$, 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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Submitted on : Wednesday, May 11, 2016 - 9:52:18 AM
Last modification on : Thursday, June 23, 2022 - 10:10:43 AM
Long-term archiving on: : Wednesday, November 16, 2016 - 12:33:36 AM

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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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