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)[source]#
Bases:
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
>>> 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: 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
>>> 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()[source]#
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
>>> 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)[source]#
Bases:
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
>>> 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: R.<x, y, z> = QQ[] sage: ProjectiveHypersurface(x - y) Projective hypersurface defined by x - y in Projective Space of dimension 2 over Rational Field
>>> 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()[source]#
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
>>> 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
- sage.schemes.generic.hypersurface.is_Hypersurface(self)[source]#
Return True if
self
is a hypersurface, i.e. an object of the typeProjectiveHypersurface
orAffineHypersurface
.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) doctest:warning... DeprecationWarning: The function is_Hypersurface is deprecated; use 'isinstance(..., (ProjectiveHypersurface, AffineHypersurface))' instead. See https://github.com/sagemath/sage/issues/38022 for details. True
>>> from sage.all import * >>> from sage.schemes.generic.hypersurface import is_Hypersurface >>> R = ZZ['x, y, z']; (x, y, z,) = R._first_ngens(3) >>> H = ProjectiveHypersurface(x*z + y**Integer(2)) >>> is_Hypersurface(H) doctest:warning... DeprecationWarning: The function is_Hypersurface is deprecated; use 'isinstance(..., (ProjectiveHypersurface, AffineHypersurface))' instead. See https://github.com/sagemath/sage/issues/38022 for details. True
sage: H = AffineHypersurface(x*z + y^2) sage: is_Hypersurface(H) True
>>> from sage.all import * >>> H = AffineHypersurface(x*z + y**Integer(2)) >>> is_Hypersurface(H) True
sage: H = ProjectiveSpace(QQ, 5) sage: is_Hypersurface(H) False
>>> from sage.all import * >>> H = ProjectiveSpace(QQ, Integer(5)) >>> is_Hypersurface(H) False