Guessing Gr{\"o}bner Bases of Structured Ideals of Relations of Sequences

TL;DR

This paper introduces algorithms that leverage structural properties of sequences and their relation ideals to efficiently guess linear recurrence relations, reducing computational effort and improving accuracy in combinatorial applications.

Contribution

It presents novel methods to incorporate structure into Gr{"o}bner basis computations for relation guessing, enabling handling of larger problems and reducing false positives.

Findings

Algorithms successfully recover relation ideals with fewer queries.

Structured approaches reduce false relations in combinatorial sequence analysis.

Experimental results demonstrate improved efficiency and capability over previous methods.

Abstract

Assuming sufficiently many terms of a n-dimensional table defined over a field are given, we aim at guessing the linear recurrence relations with either constant or polynomial coefficients they satisfy. In many applications, the table terms come along with a structure: for instance, they may be zero outside of a cone, they may be built from a Gr{\"o}bner basis of an ideal invariant under the action of a finite group. Thus, we show how to take advantage of this structure to both reduce the number of table queries and the number of operations in the base field to recover the ideal of relations of the table. In applications like in combinatorics, where all these zero terms make us guess many fake relations, this allows us to drastically reduce these wrong guesses. These algorithms have been implemented and, experimentally, they let us handle examples that we could not manage otherwise. Furthermore, we show which kind of cone and lattice structures are preserved by skew polynomial multiplication. This allows us to speed up the guessing of linear recurrence relations with polynomial coefficients by computing sparse Gr{\"o}bner bases or Gr{\"o}bner bases of an ideal invariant under the action of a finite group in a ring of skew polynomials.