Trading Order for Degree in Creative Telescoping

TL;DR

This paper studies the degrees of differential equations generated by creative telescoping for hyperexponential functions, providing formulas to estimate their complexity and improve computational efficiency.

Contribution

It derives degree bounds for creative telescoping equations, revealing the relationship between order and degree, and discusses implications for algorithm performance.

Findings

Low order equations have high degree

Higher order equations have lower degree

Degree bounds inform complexity estimates

Abstract

We analyze the differential equations produced by the method of creative telescoping applied to a hyperexponential term in two variables. We show that equations of low order have high degree, and that higher order equations have lower degree. More precisely, we derive degree bounding formulas which allow to estimate the degree of the output equations from creative telescoping as a function of the order. As an application, we show how the knowledge of these formulas can be used to improve, at least in principle, the performance of creative telescoping implementations, and we deduce bounds on the asymptotic complexity of creative telescoping for hyperexponential terms.