Algebraic independence of the partial derivatives of certain functions with arbitrary number of variables

TL;DR

This paper constructs a complex entire function with multiple variables whose all partial derivatives at algebraic points are algebraically independent, using innovative techniques involving linear isomorphisms, infinite products, and Mahler's method.

Contribution

It introduces a new method combining linear isomorphisms, infinite products, and Mahler's method to establish algebraic independence of derivatives at algebraic points.

Findings

Constructed a function with algebraically independent derivatives at algebraic points.

Developed a novel technique replacing algebraic independence with more manageable properties.

Reduced the problem to linear independence of rational functions using valuation and generic points.

Abstract

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.