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.
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.
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.
and 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 dcomplex variables of order at most ρ generating a field K 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 gave a simpler proof of Bombieri's theorem, with a slightly stronger bound of d for the degree, where the ρj are the orders of d+1 algebraically independent functions. The special cased = 1 gives the Schneider-Lang theorem, with a bound of for the number of points.
Example
If p is a polynomial with integer coefficients then the functions z1,...,zn,ep are all algebraic at a dense set of points of the hypersurface p=0.
