Submission date: 6 June 2016

Abstract:

Let (K;+,⋅, ′, 0, 1) be a differentially closed field. In this paper we explore the connection between Ax-Schanuel type theorems (predimension inequalities) for a differential equation E(x,y) and the geometry of the set U:={ y:E(t,y) ∧ y' ≠ 0 } where t is an element with t'=1. We show that certain types of predimension inequalities will imply strong minimality and geometric triviality of U. Furthermore, if E has some special form then we prove that all fibres U_s:={y:E(s,y)∧ y' ≠ 0} (with s non-constant) have the same properties. A characterisation of the induced structure on U_s^n via special subvarieties is also obtained. In particular, since an Ax-Schanuel theorem for the j-function is known (due to Pila and Tsimerman), our results will give another proof for a theorem of Freitag and Scanlon stating that the differential equation of j defines a strongly minimal set with trivial geometry (which is not ℵ_0-categorical though).

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Full text arXiv 1606.01778: pdf, ps.