Not Pfaffian

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. Why Pfaffian?Pfaffian functions were introduced in [17] where Khovanskii showed that the class has strong finiteness properties. For instance, Khovanskii exhibits an effective bound on the number of zeros in a system of equations involving Pfaffian functions. Later, the strong finiteness properties of this class of functions played an important role in model completeness results for o-minimal expansions of the real field [37,25]. Any algebraic function is Pfaffian on a suitable domain, but there are two well-known ways to see that a transcendental function is not Pfaffian:• The function is differentially transcendental. 1• The function violates the strong finiteness properties of the class. 2 This manuscript introduces a third way of showing a function is not Pfaffian. Our work is based on a very simple idea -Pfaffian functions are built using solutions to certain order one differential equations, while solutions to higher order strongly minimal differential equations can not satisfy order one differential equations. For the notion of C-Pfaffian, this summary gives nearly a complete indication of the proof that a solution to a higher order strongly minimal equation is not C-Pfaffian. There are only slightly more complications to describe when considering if the real and imaginary parts of the function are Pfaffian. These complications can already be seen in the work of Macintyre [24], who shows that Date: September 21, 2021. J. Freitag is partially supported by NSF CAREER award 1945251.1 Sometimes this property is also called being hypertranscendental or transcendentally transcendental [35]. The Gamma-function is differentially transcendental by a classical theorem of Hölder [16] and its restriction to (0, ∞) is definable in an o-minimal expansion of the reals [12].

On the equations of Poizat and Liénard

“…Both notions are closely connected to model theoretic notions from the theory of differentially closed fields. See [20].…”

Strong minimality of triangle functions

“…From there, we use Nishioka's theorem 1.2 with a model theoretic study of fibrations inspired by the work of [16] to prove the strong minimality of Schwarz triangle functions. The motivation for establishing the strong minimality of certain differential equations has been discussed many places and ranges from functional transcendence theorems [3] to diophantine results [9,10,11] to understanding solutions of equations from special classes of functions [5,6].…”