Schneider–Lang theorem
In mathematics, the Schneider–Lang theorem is a refinement by Lang (1966) of a theorem of Schneider (1949) about the transcendence of values of meromorphic functions. The theorem implies both the Hermite–Lindemann and Gelfond–Schneider theorems, and implies the transcendence of some values of elliptic functions and elliptic modular functions.
The theorem deals with a number field K and meromorphic functions f1, ..., fN, at least two of which are algebraically independent of orders ρ1 and ρ2, and such that if we differentiate any of these functions then the result is a polynomial in f1, ..., fN with coefficients in K. Under these hypotheses the theorem states that if there are m distinct complex numbers ω1, ..., ωm such that fi (ωj ) is in K for all combinations of i and j, then m is bounded by
To prove the result Lang took two algebraically independent functions from f1, ..., fN, say f and g, and then created an auxiliary function which was simply a polynomial F in f and g. This auxiliary function could not be explicitly stated since f and g are not explicitly known. But using Siegel's lemma Lang showed how to make F in such a way that it vanished to a high order at the m complex numbers ω1,...,ωm. Because of this high order vanishing it can be shown that a high-order derivative of F takes a value of small size one of the ωis, "size" here referring to an algebraic property of a number. Using the maximum modulus principle Lang also found a separate way to estimate the absolute values of derivatives of F, and using standard results comparing the size of a number and its absolute value he showed that these estimates were contradicted unless the claimed bound on m holds.
...
Wikipedia