Special classes of functions

TL;DR

This paper uses model theory and differential algebra to analyze algebraic differential equations, providing conditions for Pfaffian solutions and introducing concepts like d-irreducibility.

Contribution

It offers new necessary and sufficient conditions for solutions of algebraic differential equations and addresses open questions in the field.

Findings

Identifies conditions for algebraic ODEs to have complex Pfaffian solutions.

Provides examples of algebraic ODEs without real Pfaffian solutions.

Introduces a sufficient condition for d-irreducibility of functions.

Abstract

Using model theory and differential algebra, we give necessary conditions for algebraic ordinary differential equations to have a complex Pfaffian solution on some complex domain. These tools also allow us to give many examples of algebraic ordinary differential equations that do not have real Pfaffian solution on any open interval. We also give a sufficient condition for a function to be d-irreducible, in the sense of Nishioka. These characterizations are used to give several answers to questions of Bianconi (2016) and strengthen a theorem of Nguyen (2009).