Some observations about the real and imaginary parts of complex Pfaffian functions

Summary

Introduction

The main result of this paper has a simple proof, but is a central component in a large-scale project I have recently completed on the model theory of elliptic functions [7, 8]. In that project I take up important work of Bianconi [2] from around 1990, on the Weierstrass ℘ functions on an appropriate domain, and carry it to a decidability result modulo André's conjecture on 1-motives [1]. Bianconi proved model-completeness results, nonconstructively, for the basic situation, and I can see no way to constructivize the method he uses. Instead, I use ideas from two major developments subsequent to Bianconi's work, namely the work of Wilkie [10] and Macintyre-Wilkie [9], and the work of Gabrielov [4] and Gabrielov-Vorobjov [5]. To link with these papers, I interpret Bianconi's formulations in one based on taking the compositional inverse ℘⁻¹, on an appropriate compact, as primitive. The latter function, in contrast to ℘, is complex Pfaffian, and this alone yields, by a result of Gabrielov [5] an important constructive multiplicity bound. But I need more, namely that the real and imaginary parts of ℘⁻¹ are real Pfaffian and this is what I prove below. I doubt that there is any nontrivial general result allowing one to deduce that the real and imaginary parts of a complex Pfaffian function are real Pfaffian, and it is for this reason that I choose to publish the simple, useful result below.