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We construct a complex entire function with arbitrary number of variableswhich has the following property: The infinite set consisting of all the valuesof all its partial derivatives of any orders at all algebraic points, includingzero components, is algebraically independent. In Section 2 of this paper, wedevelop a new technique involving linear isomorphisms and infinite products toreplace the algebraic independence of the values of functions in question withthat of functions easier to deal with. In Section 2 and 3, using the techniquetogether with Mahler's method, we can reduce the algebraic independence of theinfinite set mentioned above to the linear independence of certain rationalfunctions modulo the rational function field of many variables. The latter oneis solved by the discussions involving a certain valuation and a generic pointin Section 3 and 4.

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