Sunday 7 July 2019

Dimension of Words

Dimension of Words
[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 ' is the number of different each other,

dim W ≥ l ' -r({Di}) + q(W) .
(ii) {Di} is supposed to be rich divisor at general location.
(l - n) dim W ≤n(r({Di}) - q(W)) .
[Interpretation of Theorem ( Noguchi, Winkelmann)]
k is language.
V is word.
W is meaning.
Di is meaning minimum.
has dimension that is defined at sup. or inf.

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