Dimension of Words

Diophantine Language

Dimension of Words

TANAKA Akio

[Preparation 1]
k is algebraic field.
V is non-singular projective algebraic manifold over k.
D is reduced divisor over k.
Logarithmic irregular index, q (V \ D) =  is supposed.

[Theorem, Vojta 1996]
Under Preparation 1, for (S,D)-integar subset Z  V (k) \ D,
there exists Zariski closed proper subset and there becomes 

[Preparation 2]
k is algebraic field.
V is n-dimensional projective algebraic manifold.
 are different reduced divisors each other over V.
 .
W is (S,D)-integar subset Z  V (k) \ D 's Zariski closuere in V.

[Theorem, Noguchi・Winkelmann, 2002]
(i) When l ' is the number of different each other,

dim W ≥ l ' -r({D_i}) + q(W) .
(ii) {D_i} is supposed to be rich divisor at general location.
(l - n) dim W ≤n(r({D_i}) - q(W)) ⁺ .

[Interpretation of Theorem ( Noguchi, Winkelmann)]
k is language.
V is word.
W is meaning.
D_i is meaning minimum.
W has dimension that is defined at sup. or inf.

[References]
Holomorphic Meaning Theory 2 / Tokyo June 19, 2008 Energy Distance Theory / Conjecture 1 / Word and Meaning Minimum / Tokyo September 22, 2008
Language, Word, Distance, Meaning and Meaning Minimum by Riemann-Roch Formula / Tokyo August 15, 2009

Tokyo
January 30, 2012Sekinan Research Field of Language