Let $\mathcal{K}:=(K;+,\cdot ,D,0,1)$ be a differentially closed field of characteristic 0 with field of constants C.

In the first part of the paper we explore the connection between Ax-Schanuel type theorems (predimension inequalities) for a differential equation $E(x,y)$ and the geometry of the fibres $U_s:=\{y:E(s,y)\wedge y\notin C\}$ where s is a non-constant element. We show that certain types of predimension inequalities imply strong minimality and geometric triviality of $U_s$. Moreover, the induced structure on the Cartesian powers of $U_s$ is given by special subvarieties. In particular, since the j-function satisfies an Ax-Schanuel inequality of the required form (due to Pila and Tsimerman), applying our results to the j-function we recover a theorem of Freitag and Scanlon stating that the differential equation of j defines a strongly minimal set with trivial geometry.

In the second part of the paper we study strongly minimal sets in the j-reducts of differentially closed fields. Let $E_j(x,y)$ be the (two-variable) differential equation of the j-function. We prove a Zilber style classification result for strongly minimal sets in the reduct $\mathsf{K}:=(K;+,\cdot ,E_j)$. More precisely, we show that in $\mathsf{K}$ all strongly minimal sets are geometrically trivial or non-orthogonal to C. Our proof is based on the Ax-Schanuel theorem and a matching Existential Closedness statement which asserts that systems of equations in terms of $E_j$ have solutions in $\mathsf{K}$ unless having a solution contradicts Ax-Schanuel.

让 $\mathcal{ķ}：=（ķ;+，\cdot ，d，0，\mathrm{1个}）$是特征为0且常数为C的微分封闭域。

在本文的第一部分中，我们探讨了微分方程的Ax-Schanuel型定理（维度不等式）之间的联系 $Ë（X，ÿ）$ 和纤维的几何形状 ${ü}_s：=\{ÿ：Ë（s，ÿ）\wedge ÿ\notin C\}$其中s是一个非常数元素。我们表明，某些类型的维度不等式暗示了极小的极小值和几何平凡性${ü}_s$。而且，归纳结构的笛卡儿幂${ü}_s$由特殊子变量给出。特别地，由于j函数满足所需形式的Ax-Schanuel不等式（由于Pila和Tsimerman），因此将我们的结果应用于j函数，我们恢复了Freitag和Scanlon定理，指出j的微分方程定义了具有平凡几何的极小集合。

在本文的第二部分中，我们研究了差分闭合域的j-约简中的极小集。让${Ë}_{Ĵ}（X，ÿ）$是j函数的（二变量）微分方程。我们证明了Zilber样式分类结果在归约中的极小集$\mathsf{ķ}：=（ķ;+，\cdot ，{Ë}_{Ĵ}）$。更确切地说，我们在$\mathsf{ķ}$对于C，所有极小集在几何上都是平凡的或非正交的。我们的证明基于Ax-Schanuel定理和匹配的Existential Closedness陈述，该陈述断言以${Ë}_{Ĵ}$ 有解决方案 $\mathsf{ķ}$ 除非有解决方案与Ax-Schanuel矛盾。