Diophantine Language Finiteness of Words 2012

Diophantine Language

Finiteness of Words

TANAKA Akio

[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.
A is language.

[References]
From Cell to Manifold / Cell Theory / Tokyo June 2, 2007
Amplitude of Meaning Minimum / Complex Manifold Deformation Theory / Tokyo December 17, 2008
Language, Word, Distance, Meaning and Meaning Minimum by Riemann-Roch Formula / Tokyo August 15, 2009

Tokyo
January 29, 2012
Sekinan Research Field of Language