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

Abstract

Let $(K;+,\cdot, ', 0, 1)$ 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 $U:=\{ y:E(t,y) \wedge y' \neq 0 \}$ 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 $U_s:=\{ y:E(s,y) \wedge y' \neq 0 \}$ (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).