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Schneider–Lang theorem

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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.

Contents

Statement

The theorem deals with a number field K and meromorphic functions f1, ..., fN, 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, ..., fN with coefficients in K. Under these hypotheses the theorem states that if there are m distinct complex numbers ω1, ..., ωm such that fi (ωj ) is in K for all combinations of i and j, then m is bounded by

m ≤ ( ρ 1 + ρ 2 ) [ K : Q ] .

Examples

  • If the two functions are f1 = z and f2 = ez then the theorem implies the Hermite–Lindemann theorem that eα 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 = ez and f2 = eβz for β irrational algebraic implies the Gelfond–Schneider theorem that αβ 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
    • [ ℘ ′ ( z ) ] 2 = 4 [ ℘ ( z ) ] 3 − g 2 ℘ ( z ) − g 3 , is transcendental for any algebraic α.
  • Taking the functions to be z and ef(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 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,...,ωm. 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 ωis, "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, ..., 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 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+...+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 p is a polynomial with integer coefficients then the functions z1,...,zn,ep(z1,...,zn) are all algebraic at a dense set of points of the hypersurface p=0.

    References

    Schneider–Lang theorem Wikipedia
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