Schneider–Lang theorem

Lang (1966) proved a result using this non-explicit form of auxiliary functions. The theorem implies both the Hermite–Lindemann and Gelfond–Schneider theorems. The theorem deals with a number field K and meromorphic functions f₁,...,f_N of order at most ρ, at least two of which are algebraically independent, and such that if we differentiate any of these functions then the result is a polynomial in all of the functions. Under these hypotheses the theorem states that if there are m distinct complex numbers ω₁,...,ω_m such that f_i (ω_j ) is in K for all combinations of i and j, then m is bounded by

$m\leq 20\rho [K:\mathbb {Q} ].$

To prove the result Lang took two algebraically independent functions from f₁,...,f_N, 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,...,ω_m. 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 ω_is, "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.

References

• S. Lang, "Introduction to Transcendental Numbers," Addison–Wesley Publishing Company, (1966)
• Bombieri, Enrico; Lang, Serge (1970), "Analytic subgroups of group varieties", Inventiones Mathematicae, 11: 1–14, doi:10.1007/BF01389801, ISSN 0020-9910, MR0296028
• Schneider, Theodor (1949), "Ein Satz über ganzwertige Funktionen als Prinzip für Transzendenzbeweise", Mathematische Annalen, 121: 131–140, doi:10.1007/BF01329621, ISSN 0025-5831, MR0031498