Ax-Schanuel type theorems and geometry of strongly minimal sets in differentially closed fields

2016/06/06 by Aslanyan, Vahagn
#03C60 #11F03 #12H05 #12H20 #FOS: Mathematics #Logic (math.LO)

paper · doi:10.48550/arxiv.1606.01778

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) \wedge y' ≠ 0 \ where t is an element with t'=1. We show that certain types of predimension inequalities imply strong minimality and geometric triviality of U. Moreover, the induced structure on Cartesian powers of U is given by special subvarieties. If E has some special form then all fibres U_s:=\ y:E(s,y) \wedge y' ≠ 0 \ (with s non-constant) have the same properties. In particular, since the j-function satisfies an Ax-Schanuel theorem of the required form (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 ℵ₀-categorical though).

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