Hypersurfaces in affine and projective space - Schemes

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# Hypersurfaces in affine and projective space

AUTHORS:

- William Stein <wstein@gmail.com> (2005-12-08)

- David Kohel <kohel@maths.usyd.edu.au> (2005-12-08)

- Alex Ghitza <aghitza@alum.mit.edu> (2009-04-17)

class sage.schemes.generic.hypersurface.AffineHypersurface(poly, ambient=None)

  Bases: `AlgebraicScheme_subscheme_affine`

  The affine hypersurface defined by the given polynomial.

  EXAMPLES:

  Sage

      sage: A.<x, y, z> = AffineSpace(ZZ, 3)
      sage: AffineHypersurface(x*y - z^3, A)
      Affine hypersurface defined by -z^3 + x*y
       in Affine Space of dimension 3 over Integer Ring

  Python

      >>> from sage.all import *
      >>> A = AffineSpace(ZZ, Integer(3), names=('x', 'y', 'z',)); (x, y, z,) = A._first_ngens(3)
      >>> AffineHypersurface(x*y - z**Integer(3), A)
      Affine hypersurface defined by -z^3 + x*y
       in Affine Space of dimension 3 over Integer Ring

  Sage

      sage: A.<x, y, z> = QQ[]
      sage: AffineHypersurface(x*y - z^3)
      Affine hypersurface defined by -z^3 + x*y
       in Affine Space of dimension 3 over Rational Field

  [Python]

      >>> from sage.all import *
      >>> A = QQ['x, y, z']; (x, y, z,) = A._first_ngens(3)
      >>> AffineHypersurface(x*y - z**Integer(3))
      Affine hypersurface defined by -z^3 + x*y
       in Affine Space of dimension 3 over Rational Field

  defining_polynomial()

    Return the polynomial equation that cuts out this affine hypersurface.

    EXAMPLES:

    Sage

        sage: R.<x, y, z> = ZZ[]
        sage: H = AffineHypersurface(x*z + y^2)
        sage: H.defining_polynomial()
        y^2 + x*z

    Python

        >>> from sage.all import *
        >>> R = ZZ['x, y, z']; (x, y, z,) = R._first_ngens(3)
        >>> H = AffineHypersurface(x*z + y**Integer(2))
        >>> H.defining_polynomial()
        y^2 + x*z

class sage.schemes.generic.hypersurface.ProjectiveHypersurface(poly, ambient=None)

  Bases: `AlgebraicScheme_subscheme_projective`

  The projective hypersurface defined by the given polynomial.

  EXAMPLES:

  Sage

      sage: P.<x, y, z> = ProjectiveSpace(ZZ, 2)
      sage: ProjectiveHypersurface(x - y, P)
      Projective hypersurface defined by x - y
       in Projective Space of dimension 2 over Integer Ring

  Python

      >>> from sage.all import *
      >>> P = ProjectiveSpace(ZZ, Integer(2), names=('x', 'y', 'z',)); (x, y, z,) = P._first_ngens(3)
      >>> ProjectiveHypersurface(x - y, P)
      Projective hypersurface defined by x - y
       in Projective Space of dimension 2 over Integer Ring

  Sage

      sage: R.<x, y, z> = QQ[]
      sage: ProjectiveHypersurface(x - y)
      Projective hypersurface defined by x - y
       in Projective Space of dimension 2 over Rational Field

  [Python]

      >>> from sage.all import *
      >>> R = QQ['x, y, z']; (x, y, z,) = R._first_ngens(3)
      >>> ProjectiveHypersurface(x - y)
      Projective hypersurface defined by x - y
       in Projective Space of dimension 2 over Rational Field

  defining_polynomial()

    Return the polynomial equation that cuts out this projective hypersurface.

    EXAMPLES:

    Sage

        sage: R.<x, y, z> = ZZ[]
        sage: H = ProjectiveHypersurface(x*z + y^2)
        sage: H.defining_polynomial()
        y^2 + x*z

    Python

        >>> from sage.all import *
        >>> R = ZZ['x, y, z']; (x, y, z,) = R._first_ngens(3)
        >>> H = ProjectiveHypersurface(x*z + y**Integer(2))
        >>> H.defining_polynomial()
        y^2 + x*z