A Family of Denominator Bounds for First Order Linear Recurrence Systems

TL;DR

This paper introduces a family of denominator bounds for first order linear recurrence systems, balancing bound sharpness and computational efficiency, applicable to various types of recurrence systems.

Contribution

It proposes a new family of bounds that interpolate between existing bounds, optimizing the trade-off between accuracy and computational cost for solving recurrence systems.

Findings

Family of bounds includes known bounds as special cases

Intermediate bounds offer optimal balance of sharpness and efficiency

Applicable to diverse recurrence system types

Abstract

For linear recurrence systems, the problem of finding rational solutions is reduced to the problem of computing polynomial solutions by computing a content bound or a denominator bound. There are several bounds in the literature. The sharpest bound leads to polynomial solutions of lower degrees, but this advantage need not compensate for the time spent on computing that bound. To strike the best balance between sharpness of the bound versus CPU time spent obtaining it, we will give a family of bounds. The $J$'th member of this family is similar to (Abramov, Barkatou, 1998) when $J=1$, similar to (van Hoeij, 1998) when $J$ is large, and novel for intermediate values of $J$, which give the best balance between sharpness and CPU time. The setting for our content bounds are systems $\tau(Y) = MY$ where $\tau$ is an automorphism of a UFD, and $M$ is an invertible matrix with entries in its field of fractions. This setting includes the shift case, the $q$-shift case, the multi-basic case and others. We give two versions, a global version, and a version that bounds each entry separately.