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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: `sage.schemes.generic.algebraic_scheme.AlgebraicScheme_subscheme_affine`

  The affine hypersurface defined by the given polynomial.

  EXAMPLES:

      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

      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

  `defining_polynomial`()

    Return the polynomial equation that cuts out this affine hypersurface.

    EXAMPLES:

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

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

  Bases: `sage.schemes.generic.algebraic_scheme.AlgebraicScheme_subscheme_projective`

  The projective hypersurface defined by the given polynomial.

  EXAMPLES:

      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

      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

  `defining_polynomial`()

    Return the polynomial equation that cuts out this projective hypersurface.

    EXAMPLES:

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

`sage.schemes.generic.hypersurface.``is_Hypersurface`(self)

  Return True if self is a hypersurface, i.e. an object of the type ProjectiveHypersurface or AffineHypersurface.

  EXAMPLES:

      sage: from sage.schemes.generic.hypersurface import is_Hypersurface
      sage: R.<x, y, z> = ZZ[]
      sage: H = ProjectiveHypersurface(x*z+y^2)
      sage: is_Hypersurface(H)
      True

      sage: H = AffineHypersurface(x*z+y^2)
      sage: is_Hypersurface(H)
      True

      sage: H = ProjectiveSpace(QQ, 5)
      sage: is_Hypersurface(H)
      False

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