From Wikipedia, the free encyclopedia

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, ..., fN, of which at least two are algebraically independent and have orders ρ1 and ρ2, and such that fjâ€ČKf1, ..., 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

  • 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) = eÎČz for ÎČ irrational algebraic implies the Gelfond–Schneider theorem that if α and αÎČ 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), ℘â€Č(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, ..., 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

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 d(ρ1 + ... + 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

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

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; 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 algĂ©briques d'une application mĂ©romorphe (d'aprĂšs E. Bombieri) Exp. No. 384", SĂ©minaire Bourbaki, 23Ăšme annĂ©e (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 ĂŒber ganzwertige Funktionen als Prinzip fĂŒr Transzendenzbeweise", Mathematische Annalen, 121: 131–140, doi: 10.1007/BF01329621, ISSN  0025-5831, MR  0031498, S2CID  120386931
  • Waldschmidt, Michel (1979), Nombres transcendants et groupes algĂ©briques, AstĂ©risque, vol. 69, Paris: SociĂ©tĂ© MathĂ©matique de France
From Wikipedia, the free encyclopedia

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, ..., fN, of which at least two are algebraically independent and have orders ρ1 and ρ2, and such that fjâ€ČKf1, ..., 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

  • 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) = eÎČz for ÎČ irrational algebraic implies the Gelfond–Schneider theorem that if α and αÎČ 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), ℘â€Č(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, ..., 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

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 d(ρ1 + ... + 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

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

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; 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 algĂ©briques d'une application mĂ©romorphe (d'aprĂšs E. Bombieri) Exp. No. 384", SĂ©minaire Bourbaki, 23Ăšme annĂ©e (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 ĂŒber ganzwertige Funktionen als Prinzip fĂŒr Transzendenzbeweise", Mathematische Annalen, 121: 131–140, doi: 10.1007/BF01329621, ISSN  0025-5831, MR  0031498, S2CID  120386931
  • Waldschmidt, Michel (1979), Nombres transcendants et groupes algĂ©briques, AstĂ©risque, vol. 69, Paris: SociĂ©tĂ© MathĂ©matique de France

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