# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1701.07802