Affine variety command

[Neda]

Hi Is there any commands for writing affine variety (Let k be a field, and let f1,...,fs be polynomials in k[ x1,...,xn]. Then we set V(f1,...,fs)= { (a1,...,an) ? k^n : fi (a1,...,an)=0 for all 1 ?i?s } we call V(f1,...,fs) the affine variety defined by f1,...,fs ) ?

Also how can I drew an affine variety plane such as V(x^2- y^2*z^2+ z^3) ?

[Volker Braun]

On Tuesday, April 2, 2013 9:32:40 AM UTC+1, Neda wrote:

Hi Is there any commands for writing affine variety (Let k be a field, and let f1,...,fs be polynomials in k[ x1,...,xn]. Then we set V(f1,...,fs)= { (a1,...,an) ? k^n : fi (a1,...,an)=0 for all 1 ?i?s } we call V(f1,...,fs) the affine variety defined by f1,...,fs ) ?

sage: A3.<x,y,z> = AffineSpace(QQ, 3)

sage: V = A3.subscheme([x^2- y^2*z^2+ z^3])

sage: V

Closed subscheme of Affine Space of dimension 3 over Rational Field defined by:

  -y^2*z^2 + z^3 + x^2

 

Also how can I drew an affine variety plane such as V(x^2- y^2*z^2+ z^3) ?

sage: var('x,y,z')

sage: implicit_plot3d(x^2- y^2*z^2+ z^3, [-3,3], [-3,3], [-3,3]) 

 

[John Cremona]

There is an AffineHypersurface construction:

sage: A3Q.<X,Y,Z> = AffineSpace(QQ,3)
sage: f = X^2- Y^2*Z^2+ Z^3
sage: H = AffineHypersurface(f)
sage: H
Affine hypersurface defined by -Y^2*Z^2 + Z^3 + X^2 in Affine Space of

dimension 3 over Rational Field

but there is very little functionality about what to do with such a
thing currently implemented in Sage: no plotting (but see Volker's
solution for that), you can do H.rational_points(bound=3) but it is
slow.

John

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[Neda]

 I have a problem.. I don't know the subscheme command, what is commonly use for it? thank you