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
Fix a number field K and meromorphic f1, ..., fN, of which at least two are algebraically independent and have orders ρ1 and ρ2, and such that fj′ ∈ K[f1, ..., fN] for any j. Then there are at most
(
ρ
1
+
ρ
2
)
[
K
:
Q
]
{\displaystyle (\rho _{1}+\rho _{2})[K:\mathbb {Q} ]\,}
distinct complex numbers ω1, ..., 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 algebraic α: 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 β irrational 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.
Recall that the Weierstrass P function satisfies the differential equation
℘
′
(
z
)
2
=
4
℘
(
z
)
3
−
g
2
℘
(
z
)
−
g
3
.
{\displaystyle \wp '(z)^{2}=4\wp (z)^{3}-g_{2}\wp (z)-g_{3}.\,}
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
d
(
(
d
+
1
)
ρ
[
K
:
Q
]
+
1
)
.
{\displaystyle d((d+1)\rho [K:\mathbb {Q} ]+1).}
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
{\displaystyle p}
is a polynomial with integer coefficients then the functions
z
1
,
.
.
.
,
z
n
,
e
p
(
z
1
,
.
.
.
,
z
n
)
{\displaystyle z_{1},...,z_{n},e^{p(z_{1},...,z_{n})}}
are all algebraic at a dense set of points of the hypersurface
p
=
0
{\displaystyle p=0}
.