Corpus ID: 237572109

# Not Pfaffian

@inproceedings{Freitag2021NotP,
title={Not Pfaffian},
author={James Freitag},
year={2021}
}
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… Expand

#### References

SHOWING 1-10 OF 38 REFERENCES
Some Model Theory of Hypergeometric and Pfaffian Functions
We present some results and open problems related to expansions of the field of real numbers by hypergeometric and related functions focussing on definability and model completeness questions. InExpand
Pfaffian definitions of Weierstrass elliptic functions
• Mathematics
• 2017
We give explicit definitions of the Weierstrass elliptic functions $$\wp$$ ℘ and $$\zeta$$ ζ in terms of pfaffian functions, with complexity independent of the lattice involved. We also give such aExpand
Ax-Lindemann-Weierstrass with derivatives and the genus 0 Fuchsian groups
• Mathematics
• 2018
We prove the Ax-Lindemann-Weierstrass theorem with derivatives for the uniformizing functions of genus zero Fuchsian groups of the first kind. Our proof relies on differential Galois theory,Expand
Strong minimality and the j-function
• Mathematics
• 2014
We show that the order three algebraic differential equation over ${\mathbb Q}$ satisfied by the analytic $j$-function defines a non-$\aleph_0$-categorical strongly minimal set with trivial forkingExpand
Lascar and Morley Ranks Differ in Differentially Closed Fields
• Mathematics, Computer Science
• J. Symb. Log.
• 1999
We note here, in answer to a question of Poizat, that the Morley and Lascar ranks need not coincide in differentially closed fields. We will approach this through the (perhaps) more fundamental issueExpand
The field of reals with multisummable series and the exponential function
• Mathematics
• 2000
We show that the field of real numbers with multisummable real power series is model complete, o-minimal and polynomially bounded. Further expansion by the exponential function yields again a modelExpand
Pfaffian control of some polynomials involving the $j$--function and Weierstrass elliptic functions.
We give new bounds on the zeros of polynomials in $z$ and the $j$--function, and $z$ and Weierstrass elliptic functions with rectangular associated lattice, controlling the zeros of these functionsExpand
Generic differential equations are strongly minimal
• Mathematics
• 2021
In this manuscript we develop a new technique for showing that a nonlinear algebraic differential equation is strongly minimal based on the recently developed notion of the degree of nonminimality ofExpand
THE DENSITY OF ALGEBRAIC POINTS ON CERTAIN PFAFFIAN SURFACES
• Mathematics
• 2012
We prove some instances of Wilkie's conjecture on the density of rational points on sets definable in the real exponential field. In particular, we prove that this conjecture is true for surfacesExpand
Effective Pila–Wilkie bounds for unrestricted Pfaffian surfaces
• Mathematics
• 2018
We prove effective Pila--Wilkie estimates for the number of rational points of bounded height lying on certain surfaces defined by Pfaffian functions. The class of surfaces to which our resultExpand