In mathematics, the SchneiderâLang 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 HermiteâLindemann and GelfondâSchneider theorems, and implies the transcendence of some values of elliptic functions and elliptic modular functions.
Fix a number field K and meromorphic f1, ..., fN, of which at least two are algebraically independent and have orders Ï1 and Ï2, and such that fjâČ ∈ Kf1, ..., fN for any j. Then there are at most
distinct complex numbers Ï1, ..., Ïm such that fi(Ïj) ∈ K for all combinations of i and j.
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 Ï1, ..., Ïm. Thus a high-order derivative of F takes a value of small size at one such Ïis, "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 & 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 Ï 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(Ï1 + ... + Ïd+1)[K:Q] for the degree, where the Ïj are the orders of d + 1 algebraically independent functions. The special case d = 1 gives the Schneider–Lang theorem, with a bound of (Ï1 + Ï2)[K:Q] for the number of points.
If is a polynomial with integer coefficients then the functions are all algebraic at a dense set of points of the hypersurface .
In mathematics, the SchneiderâLang 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 HermiteâLindemann and GelfondâSchneider theorems, and implies the transcendence of some values of elliptic functions and elliptic modular functions.
Fix a number field K and meromorphic f1, ..., fN, of which at least two are algebraically independent and have orders Ï1 and Ï2, and such that fjâČ ∈ Kf1, ..., fN for any j. Then there are at most
distinct complex numbers Ï1, ..., Ïm such that fi(Ïj) ∈ K for all combinations of i and j.
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 Ï1, ..., Ïm. Thus a high-order derivative of F takes a value of small size at one such Ïis, "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 & 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 Ï 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(Ï1 + ... + Ïd+1)[K:Q] for the degree, where the Ïj are the orders of d + 1 algebraically independent functions. The special case d = 1 gives the Schneider–Lang theorem, with a bound of (Ï1 + Ï2)[K:Q] for the number of points.
If is a polynomial with integer coefficients then the functions are all algebraic at a dense set of points of the hypersurface .