Sunday 13 May 2018

Diophantine Language Finiteness of Words 2012

Finiteness of Words
[Preparation 1]
k is algebraic field.
 is finite subset.
V is projective algebraic manifold over k.
D is defined divisor over k.
All the sub-manifolds are over k.
Rational point is k-rational point.
[Preparation 2]
L is rich line bundle.
|L| is complete linear system.
D is divisor of |L|.
 is regular cut to D.
 is approximate function to D.
 is counting function to D.
 is rich line bundle.
When  islarge,  becomes rich.
 is basis of  .
 is embedding.
[Definition 1]
 ,
 ,
 .

[Definition 2]
Subset of rational points  \  is integer under the next condition.
(i) There exists a certain constant  .
(ii)  \  .
[Theorem, Faltings]
A is Abelian variety over k.
When D is reduced rich divisor, arbitrary integer subset  \  is always finite set.
[Interpretation]
D is meaning minimum.
 \  is word.
is language.
Tokyo
January 29, 2012 
Sekinan Research Field of Language


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