Shidlovskii Theorem

Let , ...,  for  be a set of E-functions that (1) form a solution of the system of differential equations

for  and , ...,  and (2) are algebraically independent over . Then for all , where  denotes the set of algebraic numbers with  and distinct from singularities of the differential equations, the numbers , ...,  are algebraically independent (Nesterenko 1999).

REFERENCES:

Nesterenko, Yu. V. A Course on Algebraic Independence: Lectures at IHP 1999. Unpublished manuscript. 1999.

Shidlovskii, A. B. Transcendental Numbers. New York: de Gruyter, 1989.