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

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

 

distinct complex numbers ω1, ..., ωm such that fi(ωj) ∈ K for all combinations of i and j.

Examples

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  • If f1(z) = z and f2(z) = ez then the theorem implies the Hermite–Lindemann theorem that eα 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) = ez and f2(z) = eβz for β irrational algebraic implies the Gelfond–Schneider theorem that if α and αβ 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
 
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

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To prove the result Lang took two algebraically independent functions from f1, ..., fN, say, f and g, and then created an auxiliary function FK[ f, g]. Using Siegel's lemma, he then showed that one could assume F vanished to a high order at the ω1, ..., ωm. Thus a high-order derivative of F takes a value of small size at one such ωis, "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

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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, ..., fN are meromorphic functions of d complex variables of order at most ρ generating a field K(f1, ..., fN) of transcendence degree at least d + 1 that is closed under all partial derivatives, then the set of points where all the functions fn have values in K is contained in an algebraic hypersurface in Cd of degree at most

 

Waldschmidt (1979, theorem 5.1.1) gave a simpler proof of Bombieri's theorem, with a slightly stronger bound of d1 + ... + ρd+1)[K:Q] for the degree, where the ρj 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

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If   is a polynomial with integer coefficients then the functions   are all algebraic at a dense set of points of the hypersurface  .

References

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  • 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; 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