Schneider–Lang theorem explained

In mathematics, the Schneider–Lang theorem is a refinement by of a theorem of 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 and meromorphic, of which at least two are algebraically independent and have orders and, and such that for any . Then there are at most

(\rho _{1+\rho} _{2)} [K: Q ] distinct complex numbers such that for all combinations of and .

Examples

• If and then the theorem implies the Hermite–Lindemann theorem that is transcendental for nonzero algebraic : otherwise, would be an infinite number of values at which both and are algebraic.
• Similarly taking and for irrational algebraic implies the Gelfond–Schneider theorem that if and are algebraic, then : otherwise, would be an infinite number of values at which both and 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,, shows that, for any algebraic, if and are algebraic, then is transcendental.

• Taking the functions to be and for a polynomial of degree shows that the number of points where the functions are all algebraic can grow linearly with the order .

Proof

To prove the result Lang took two algebraically independent functions from, say, and, and then created an auxiliary function . Using Siegel's lemma, he then showed that one could assume vanished to a high order at the . Thus a high-order derivative of takes a value of small size at one such s, "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 . Standard results connect the size of a number and its absolute value, and the combined estimates imply the claimed bound on .

Bombieri's theorem

and 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: Q ] +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} ^{p(z _{1,...,z} _{n)}} ^{n,e} are all algebraic at a dense set of points of the hypersurface p =0