Fix a number fieldK and meromorphicf1, ..., fN, of which at least two are algebraically independent and have ordersρ1 and ρ2, and such that fj′ ∈ K[f1, ..., fN] for any j. Then there are at most
distinct complex numbersω1, ..., 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 algebraicα: 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 βirrational 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.
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 ef(z) for a polynomialf of degreeρ 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 F ∈ K[ 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 integercoefficients then the functions are all algebraic at a dense set of points of the hypersurface .
![{\displaystyle (\rho _{1}+\rho _{2})[K:\mathbb {Q} ]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d1d356ffd0c7e1623db5395634479755b651e073)
![{\displaystyle d((d+1)\rho [K:\mathbb {Q} ]+1).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/889b3486d2b4d1f9f283ca4f436881123582bf54)
