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]
Tokyo July 26, 2007
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