Mirsky's theorem

2012-12-23T20:37:03Z

178.117.212.41: /* The theorem */

In mathematics, the '''Schneider–Lang theorem''' is a refinement by {{harvtxt| Lang|1966}} of a theorem of {{harvtxt|Schneider|1949}} about the [[transcendental number|transcendence]] of values of [[meromorphic function]]s. The theorem implies both the [[Hermite–Lindemann theorem|Hermite–Lindemann]] and [[Gelfond–Schneider theorem]]s, and implies the transcendence of some values of [[elliptic function]]s and [[elliptic modular function]]s.

== Statement ==

The theorem deals with a [[number field]] ''K'' and [[meromorphic]] functions ''f''1, ..., ''f''''N'', at least two of which are algebraically independent of [[entire function|order]]s ρ1 and ρ2, and such that if we differentiate any of these functions then the result is a polynomial in all of the functions. Under these hypotheses the theorem states that if there are ''m'' distinct [[complex number]]s ''ω''1, ..., ''ω''''m'' such that ''f''''i'' (ω''j'' ) is in ''K'' for all combinations of ''i'' and ''j'', then ''m'' is bounded by

: <math>m\leq (\rho_1+\rho_2) [K:\mathbb{Q}]. \, </math>

== Examples ==

* If the two functions are ''f''1 = ''z'' and ''f''1 = ''e''''z'' then the theorem implies the [[Hermite–Lindemann theorem]] that ''e''α is transcendental for any nonzero algebraic α, otherwise α, 2α, 3α,... would be an infinite number of values at which both ''f''1 and ''f''2 are algebraic.
* Similarly taking the two function to be ''f''1 = ''e''''z'' and ''f''1 = ''e''β''z'' for ''β'' irrational algebraic implies the [[Gelfond–Schneider theorem]] that ''α''''β'' cannot be algebraic if α is algebraic and not 0 or 1. Otherwise log ''α'', 2 log ''α'', 3 log ''α'' would be an infinite number of values at which both ''f''1 and ''f''2 are algebraic.
* Taking the three functions to be ''z'', ℘(αz), ℘'(αz) shows that if ''g''2 and ''g''3 are algebraic then the [[Weierstrass P function]] ℘(α), which satisfies the differential equation
: <math> [\wp'(z)]^2 = 4[\wp(z)]^3-g_2\wp(z)-g_3, \, </math>
: is transcendental for any algebraic α.
* Taking the functions to be ''z'' and ''e''''f''(''z'') for a polynomial ''f'' of degree ρ 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 ''f''1, ..., ''f''''N'', say ''f'' and ''g'', and then created an auxiliary function which was simply a polynomial ''F'' in ''f'' and ''g''. This auxiliary function could not be explicitly stated since ''f'' and ''g'' are not explicitly known. But using [[Siegel's lemma]] Lang showed how to make ''F'' in such a way that it vanished to a high order at the ''m'' complex numbers
ω1,...,ωm. Because of this high order vanishing it can be shown that a high-order derivative of ''F'' takes a value of small size one of the ω''i''s, "size" here referring to an algebraic property of a number. Using the [[maximum modulus principle]] Lang also found a separate way to estimate the absolute values of derivatives of ''F'', and using standard results comparing the size of a number and its absolute value he showed that these estimates were contradicted unless the claimed bound on ''m'' holds.

== Bombieri's theorem ==
{{harvtxt|Bombieri|Lang|1970}} and {{harvtxt|Bombieri|1970}} generalized the result to functions of several variables. Bombieri showed that if ''K'' is an algebraic number field and ''f''1, ..., ''f''''N'' are meromorphic functions of ''d'' complex variables of order at most ρ generating a field ''K''( ''f''1, ..., ''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)ρ[''K'':'''Q'''] + ''d''
{{harvtxt|Waldschmidt|1979|loc=theorem 5.1.1}} 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 ==

* {{Citation | last1=Bombieri | first1=Enrico | author1-link=Enrico Bombieri | title=Algebraic values of meromorphic maps | doi=10.1007/BF01418775 | mr=0306201 | year=1970 | journal=[[Inventiones Mathematicae]] | issn=0020-9910 | volume=10 | pages=267–287 | issue=4}}, {{Citation | last1=Bombieri | first1=Enrico | author1-link=Enrico Bombieri | title=Addendum to my paper: "Algebraic values of meromorphic maps" (Invent. Math. 10 (1970), 267–287) | doi= 10.1007/BF01404610 | mr=0322203 | year=1970 | journal=[[Inventiones Mathematicae]] | issn=0020-9910 | volume=11 | pages=163–166 | issue=2}}
* {{Citation | last1=Bombieri | first1=Enrico | author1-link=Enrico Bombieri | last2=Lang | first2=Serge | author2-link=Serge Lang | title=Analytic subgroups of group varieties | doi=10.1007/BF01389801 | mr=0296028 | year=1970 | journal=[[Inventiones Mathematicae]] | issn=0020-9910 | volume=11 | pages=1–14}}
* S. Lang, "''Introduction to Transcendental Numbers''," Addison–Wesley Publishing Company, (1966)
* {{Citation | last1=Lelong | first1=Pierre | title=Séminaire Bourbaki, 23ème année (1970/1971) | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Lecture Notes in Math. | doi=10.1007/BFb0058695 | mr=0414500 | year=1971 | volume=244 | chapter=Valeurs algébriques d'une application méromorphe (d'après E. Bombieri) Exp. No. 384 | pages=29–45 | isbn=978-3-540-05720-8}}
* {{Citation | last1=Schneider | first1=Theodor | title=Ein Satz über ganzwertige Funktionen als Prinzip für Transzendenzbeweise | doi=10.1007/BF01329621 | mr=0031498 | year=1949 | journal=[[Mathematische Annalen]] | issn=0025-5831 | volume=121 | pages=131–140}}
* {{Citation | last1=Waldschmidt | first1=Michel | title=Nombres transcendants et groupes algébriques | publisher=[[Société Mathématique de France]] | location=Paris | series=Astérisque | year=1979 | volume=69}}

{{DEFAULTSORT:Schneider-Lang theorem}}
[[Category:Diophantine approximation]]
[[Category:Transcendental numbers]]

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Mirsky's theorem