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

Fix a number field K and meromorphic f₁, ..., f_N, of which at least two are algebraically independent and have orders ρ₁ and ρ₂, and such that f_j′ ∈ K[f₁, ..., f_N] for any j. Then there are at most

$(\rho _{1}+\rho _{2})[K:\mathbb {Q} ]\,$

distinct complex numbers ω₁, ..., ω_m such that f_i(ω_j) ∈ K for all combinations of i and j.

Examples

• If f₁(z) = z and f₂(z) = e^z then the theorem implies the Hermite–Lindemann theorem that e^α is transcendental for nonzero algebraic α: otherwise, α, 2α, 3α, ... would be an infinite number of values at which both f₁ and f₂ are algebraic.
• Similarly taking f₁(z) = e^z and f₂(z) = e^βz for β irrational algebraic implies the Gelfond–Schneider theorem that if α and α^β are algebraic, then α ∈ {0,1}: otherwise, log(α), 2log(α), 3log(α), ... would be an infinite number of values at which both f₁ and f₂ are algebraic.
• Recall that the Weierstrass P function satisfies the differential equation

$\wp '(z)^{2}=4\wp (z)^{3}-g_{2}\wp (z)-g_{3}.\,$

Taking the three functions to be z, ℘(αz), ℘′(αz) shows that, for any algebraic α, if g₂(α) and g₃(α) are algebraic, then ℘(α) is transcendental.

• 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 F ∈ K[ f, g]. Using Siegel's lemma, he then showed that one could assume F vanished to a high order at the ω₁, ..., ω_m. Thus a high-order derivative of F takes a value of small size at one such ω_is, "size" here referring to an algebraic property of a number. Using the maximum modulus principle, Lang also found a separate estimate for absolute values of derivatives of F. Standard results connect the size of a number and its absolute value, and the combined estimates imply the claimed bound on m.

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

$d((d+1)\rho [K:\mathbb {Q} ]+1).$

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 $z_{1},...,z_{n},e^{p(z_{1},...,z_{n})}$ 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, Bibcode:1970InMat..10..267B, doi:10.1007/BF01418775, ISSN 0020-9910, MR 0306201, S2CID 123180813
  • Bombieri, Enrico (1971), "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, S2CID 121612401
• Bombieri, Enrico; Lang, Serge (1970), "Analytic subgroups of group varieties", Inventiones Mathematicae, 11: 1–14, Bibcode:1970InMat..11....1B, doi:10.1007/BF01389801, ISSN 0020-9910, MR 0296028, S2CID 122211611
• Lang, S. (1966), Introduction to Transcendental Numbers, Addison-Wesley Publishing Company
• 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, vol. 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, S2CID 120386931
• Waldschmidt, Michel (1979), Nombres transcendants et groupes algébriques, Astérisque, vol. 69, Paris: Société Mathématique de France