Not Pfaffian

TL;DR

This paper explores the relationship between differential equations, Pfaffian functions, and o-minimal structures, providing new examples of functions that are differentially algebraic but not Pfaffian, and answering open questions in the field.

Contribution

It establishes a novel connection between strong minimality of differential equations and Pfaffian functions, and provides the first known examples of such functions in o-minimal structures.

Findings

Connected strong minimality to Pfaffian properties.

Provided the first examples of differentially algebraic but non-Pfaffian functions.

Answered open questions by Binyamini, Novikov, and Bianconi.

Abstract

This short note describes the connection between strong minimality of the differential equation satisfied by an complex analytic function and the real and imaginary parts of the function being Pfaffian. This connection combined with a theorem of Freitag and Scanlon (2017) provides the answer to a question of Binyamini and Novikov (2017). We also answer a question of Bianconi (2016). We give what seem to be the first examples of functions which are definable in o-minimal expansions of the reals and are differentially algebraic, but not Pfaffian.