论文标题
某些功能的部分衍生物具有任意数量变量的代数独立性
Algebraic independence of the partial derivatives of certain functions with arbitrary number of variables
论文作者
论文摘要
我们构建具有任意数量的变量的复杂整个函数，该函数具有以下属性：无限集由所有代数点（包括零组件）在所有代数点（包括零组件）上所有订单的所有部分衍生物组成，在代数上是独立的。在本文的第2节中，我们开发了一项涉及线性同构和无限产品的新技术，以替代所讨论的功能值的代数独立性，而功能易于处理。在第2和第3节中，使用该技术与Mahler的方法一起，我们可以将上述无限集的代数独立性降低到某些有理函数的线性独立性模型的许多变量的合理函数字段。后一个通过涉及一定估值和第3节和第4节中的通用点的讨论来解决。
We construct a complex entire function with arbitrary number of variables which has the following property: The infinite set consisting of all the values of all its partial derivatives of any orders at all algebraic points, including zero components, is algebraically independent. In Section 2 of this paper, we develop a new technique involving linear isomorphisms and infinite products to replace the algebraic independence of the values of functions in question with that of functions easier to deal with. In Section 2 and 3, using the technique together with Mahler's method, we can reduce the algebraic independence of the infinite set mentioned above to the linear independence of certain rational functions modulo the rational function field of many variables. The latter one is solved by the discussions involving a certain valuation and a generic point in Section 3 and 4.
