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.

Statement

The theorem deals with a number field K and meromorphic functions f₁, ..., f_N, 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, ..., f_N with coefficients in K. 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

$m\leq (\rho_1+\rho_2) [K:\mathbb{Q}]. \,$

Examples

• If the two functions are f₁ = z and f₂ = e^z then the theorem implies the Hermite–Lindemann theorem that e^α is transcendental for any nonzero algebraic α, otherwise α, 2α, 3α,... would be an infinite number of values at which both f₁ and f₂ are algebraic.
• Similarly taking the two function to be f₁ = e^z and f₂ = e^βz for β irrational algebraic implies the Gelfond–Schneider theorem that α^β cannot be algebraic if α is algebraic and not 0 or 1. Otherwise log α, 2 log α, 3 log α would be an infinite number of values at which both f₁ and f₂ are algebraic.
• Taking the three functions to be z, ℘(αz), ℘'(αz) shows that if g₂ and g₃ are algebraic then the Weierstrass P function ℘(α), which satisfies the differential equation

$[\wp'(z)]^2 = 4[\wp(z)]^3-g_2\wp(z)-g_3, \,$

is transcendental for any algebraic α.

• Taking the functions to be z and e^f(z) for a polynomial f of degree ρ shows that the number of points where the functions are all algebraic can grow linearly with the order ρ = deg(f).

Proof

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.

Bombieri's theorem

Bombieri & Lang (1970) and Bombieri (1970) generalized the result to functions of several variables. Bombieri showed that if K is an algebraic number field and f₁, ..., f_N are meromorphic functions of d complex variables of order at most ρ generating a field K( f₁, ..., f_N) of transcendence degree at least d + 1 that is closed under all partial derivatives, then the set of points where all the functions f_n have values in K is contained in an algebraic hypersurface in C^d of degree at most d(d + 1)ρ[K:Q] + d Waldschmidt (1979, theorem 5.1.1) gave a simpler proof of Bombieri's theorem, with a slightly stronger bound of d(ρ₁+...+ρ_d+1)[K:Q] for the degree, where the ρ_j are the orders of d+1 algebraically independent functions. The special case d = 1 gives the Schneider–Lang theorem, with a bound of (ρ₁+ρ₂)[K:Q] for the number of points.

Example. If p is a polynomial with integer coefficients then the functions z₁,...,z_n,e^{p(z₁,...,z_n)} are all algebraic at a dense set of points of the hypersurface p=0.

References

• Bombieri, Enrico (1970), "Algebraic values of meromorphic maps", Inventiones Mathematicae 10 (4): 267–287, doi:10.1007/BF01418775, ISSN 0020-9910, MR 0306201, Bombieri, Enrico (1970), "Addendum to my paper: "Algebraic values of meromorphic maps" (Invent. Math. 10 (1970), 267–287)", Inventiones Mathematicae 11 (2): 163–166, doi:10.1007/BF01404610, ISSN 0020-9910, MR 0322203
• Bombieri, Enrico; Lang, Serge (1970), "Analytic subgroups of group varieties", Inventiones Mathematicae 11: 1–14, doi:10.1007/BF01389801, ISSN 0020-9910, MR 0296028
• S. Lang, "Introduction to Transcendental Numbers," Addison–Wesley Publishing Company, (1966)
• Lelong, Pierre (1971), "Valeurs algébriques d'une application méromorphe (d'après E. Bombieri) Exp. No. 384", Séminaire Bourbaki, 23ème année (1970/1971), Lecture Notes in Math. 244, Berlin, New York: Springer-Verlag, pp. 29–45, doi:10.1007/BFb0058695, ISBN 978-3-540-05720-8, MR 0414500
• Schneider, Theodor (1949), "Ein Satz über ganzwertige Funktionen als Prinzip für Transzendenzbeweise", Mathematische Annalen 121: 131–140, doi:10.1007/BF01329621, ISSN 0025-5831, MR 0031498
• Waldschmidt, Michel (1979), Nombres transcendants et groupes algébriques, Astérisque 69, Paris: Société Mathématique de France