Annals of Pure and Applied Logic ( IF 0.6 ) Pub Date : 2020-08-04 , DOI: 10.1016/j.apal.2020.102871 Vahagn Aslanyan
Let 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 and the geometry of the fibres where s is a non-constant element. We show that certain types of predimension inequalities imply strong minimality and geometric triviality of . Moreover, the induced structure on the Cartesian powers of 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 be the (two-variable) differential equation of the j-function. We prove a Zilber style classification result for strongly minimal sets in the reduct . More precisely, we show that in 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 have solutions in unless having a solution contradicts Ax-Schanuel.
中文翻译:
Ax-Schanuel和j函数的极小值
让 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 和纤维的几何形状 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。而且,归纳结构的笛卡儿幂由特殊子变量给出。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。更确切地说,我们在对于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 有解决方案 除非有解决方案与Ax-Schanuel矛盾。