7
Projective Space
1
Closed field k
Affine space An+1
Coordinates of affine space ( X0, X1, … , Xn )
Set that has not 0 in An+1 An+1 \ { 0, 0, … , 0 }
Two elements of the set P = ( a0, a1, … , an ) Q= ( b0, b1, … , bn )
Element of k that is not 0 λ
( b0, b1, … , bn ) = ( λa0, λa1, … , λan )
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, … , Xn |
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
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