Abstract
Given several n-dimensional sequences, we first present an algorithm for computing the Grobner basis of their module of linear recurrence relations.A P-recursive sequence (ui)i ∈ Nn 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 ((ij, ui)i ∈ Nn)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 Grobner bases computations in an Ore algebra K t1,...,tn,x1,...,xn, with commutators xk,xl-xl,xk=tk,tl-tl,tk= tk,xl-xl,tk=0 for k ≠ l and tk,xk-xk,tk=xk, into the algorithm designed for P-recursive sequences. This allows us to compute faster the elements of the Grobner basis of which are in the ideal spanned by the first relations, such as in 2D/3D-space walks examples.
Highlights
Computing linear recurrence relations of multi-dimensional sequences is a fundamental problem in Computer Science
For one-dimensional sequences, C-relations can be guessed with Berlekamp – Massey algorithm [4, 20], while for multidimensional sequences, they can be guessed with Berlekamp – Massey – Sakata algorithm [24, 26] or Scalar-FGLM [5, 6]
This algorithm input is a P-recursive table, a degree bound on the order of the relations and a degree bound on the polynomial coefficients
Summary
Computing linear recurrence relations of multi-dimensional sequences is a fundamental problem in Computer Science. As this is the core of the paper, we start with an example. (i1 − ) i2)!) (i1,i2)∈N2 satisfies Pascal’s rule, a linear recurrence relation:. Thanks to equations (2a) and (2b), we can compute any term of b starting with b0,0 = 1. A P-recursive sequence is uniquely defined by the linear recurrence relations with polynomial coefficients it satisfies and a finite number of initial terms
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