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

Abstract

Let $(K;+,\cdot {,}^{\prime },0,1)$ 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 $U:=\{y:E(t,y)\wedge y^{\prime }\ne 0\}$ where $t$ is an element with $t^{\prime }=1$. 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 $U_s:=\{y:E(s,y)\wedge y^{\prime }\ne 0\}$ (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 ${\aleph }_0$-categorical though).

12 pages. Incorporated into arXiv:1805.03985v2

References in corpus (2)

• Ax-Schanuel for the j-function
• Ax-Schanuel for Linear Differential Equations

Cited by in corpus (1)

• Ax-Schanuel and strong minimality for the $j$-function