  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.

Number Theory