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D.9.1.10 closed_points

Procedure from library brnoeth.lib (see brnoeth_lib).

Usage:

closed_points(I); I an ideal

Return:

list of prime ideals (each a Groebner basis), corresponding to the (distinct affine closed) points of V(I)

Note:

The ideal must have dimension 0, the basering must have 2 variables, the ordering must be lp, and the base field must be finite and prime.
It might be convenient to set the option(redSB) in advance.

Example:

LIB "brnoeth.lib";
ring s=2,(x,y),lp;
// this is just the affine plane over F_4 :
ideal I=x4+x,y4+y;
list L=closed_points(I);
// and here you have all the points :
L;
→ [1]:
→    _[1]=y2+y+1
→    _[2]=x+y
→ [2]:
→    _[1]=y2+y+1
→    _[2]=x+1
→ [3]:
→    _[1]=y2+y+1
→    _[2]=x+y+1
→ [4]:
→    _[1]=y2+y+1
→    _[2]=x
→ [5]:
→    _[1]=y+1
→    _[2]=x2+x+1
→ [6]:
→    _[1]=y+1
→    _[2]=x+1
→ [7]:
→    _[1]=y+1
→    _[2]=x
→ [8]:
→    _[1]=y
→    _[2]=x2+x+1
→ [9]:
→    _[1]=y
→    _[2]=x+1
→ [10]:
→    _[1]=y
→    _[2]=x

See also: triang_lib.


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