Bounds for Substituting Algebraic Functions into D-finite Functions

TL;DR

This paper provides new estimates for the order and degree of annihilating operators when composing D-finite functions with algebraic functions, improving understanding of their algebraic structure.

Contribution

It introduces the first estimates for the orders and degrees of annihilators in compositions of D-finite and algebraic functions, utilizing singularity analysis for accuracy.

Findings

Order-degree curve is more accurate when analyzing removable singularities.

Provides the first quantitative bounds for such compositions.

Enhances understanding of the algebraic properties of composed functions.

Abstract

It is well known that the composition of a D-finite function with an algebraic function is again D-finite. We give the first estimates for the orders and the degrees of annihilating operators for the compositions. We find that the analysis of removable singularities leads to an order-degree curve which is much more accurate than the order-degree curve obtained from the usual linear algebra reasoning.