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

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′ 6= 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 Us := {y : E(s, y) ∧ y′ 6= 0} (with s non-constant) have the same properties. 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 א0-categorical though).