Linguistic Note 7 Projective Space / July 26, 2007

Linguistic Note

7

Projective Space

　　TANAKA Akio

1

Closed field     k

Affine space     Aⁿ⁺¹

Coordinates of affine space     ( X₀, X₁, … , X_n )

Set that has not 0 in Aⁿ⁺¹     Aⁿ⁺¹ \ { 0, 0, … , 0 }

Two elements of the set     P = ( a₀, a₁, … , a_n )        Q= ( b₀, b₁, … , b_n )

Element of k that is not 0     λ

( b₀, b₁, … , b_n ) = ( λa₀, λa₁, … , λa_n )  

P and Q are equivalent.     P ~ Q

Set of the equivalent class     Pⁿ = Aⁿ⁺¹ \ {0} / ~

n-dimensional projective space    Pⁿ

2

Polynomial ring that has n + 1 variant     S = k | X₀, X₁, … , X_n |

S’ homogeneous polynomial     T

Z (f ) = { P ∈ Pⁿ | f ( P ) = 0 }

Z (T ) = { P ∈ Pⁿ | f ( P ) = 0, ∀f ∈ T }

Subset of Pⁿ     X

X has set T that consists of S’ homogeneous polynomial.

X = Z ( T )

X is algebraic set.

3

Pⁿ that has topology which is closed set of algebraic set.        Zariski topology

4

Irreducible algebraic set of Pⁿ     Projective algebraic variety

f ∈ S     degree   d   homogeneous polynomial

Z (f ) is d degree hypersurface of Pⁿ

[Note]

Surface on which quantum exists may be described by algebra, especially for Aurora Theory and Aurora Time Theory.

[References]

Aurora Theory     Aurora Plane     Tokyo October 14, 2006

Aurora Theory     Distance and Time     Tokyo October 28, 2006

Aurora Time Theory     Imaginary Time and Imaginary Space     Tokyo November 11, 2006

Aurora Time Theory     Enlarged Distance Theory     Tokyo November 20, 2006

Aurora Time Theory     Opened Time and Closed Time     Revised     Tokyo November 25, 2006

Tokyo July 26, 2007

Sekinan Research Field of Language