Schneider–Lang theorem

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

$({\rho }_1+{\rho }_2)[K:\mathbb{Q}]$

distinct complex numbers Template:Math such that Template:Math for all combinations of Template:Mvar and Template:Mvar.

Examples

• If Template:Math and Template:Math then the theorem implies the Hermite–Lindemann theorem that Template:Math is transcendental for nonzero algebraic Template:Mvar: otherwise, Template:Math would be an infinite number of values at which both Template:Math and Template:Math are algebraic.
• Similarly taking Template:Math and Template:Math for Template:Mvar irrational algebraic implies the Gelfond–Schneider theorem that if Template:Mvar and Template:Math are algebraic, then Template:Math}: otherwise, Template:Math would be an infinite number of values at which both Template:Math and Template:Math are algebraic.
• Recall that the Weierstrass P function satisfies the differential equation

${\mathrm{\wp }}^{\prime }(z{)}^2=4\mathrm{\wp }(z{)}^3-g_2\mathrm{\wp }(z)-g_3.$

Taking the three functions to be Template:Math, Template:Math, Template:Math shows that, for any algebraic Template:Mvar, if Template:Math and Template:Math are algebraic, then Template:Math is transcendental.

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

Proof

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

Bombieri's theorem

Template:Harvtxt and Template:Harvtxt 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

$d((d+1)\rho [K:\mathbb{Q}]+1).$

Template:Harvtxt 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 $z_1,...,z_n,e^{p(z_1,...,z_n)}$ are all algebraic at a dense set of points of the hypersurface $p=0$.

References

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