def subscheme(self, X, **kwds):
"""
Return the closed subscheme defined by ``X``.
INPUT:
- ``X`` - a list or tuple of equations.
EXAMPLES::
sage: A.<x,y> = AffineSpace(QQ, 2)
sage: X = A.subscheme([x, y^2, x*y^2]); X
Closed subscheme of Affine Space of dimension 2 over Rational Field defined by:
x,
y^2,
x*y^2
::
sage: X.defining_polynomials ()
(x, y^2, x*y^2)
sage: I = X.defining_ideal(); I
Ideal (x, y^2, x*y^2) of Multivariate Polynomial Ring in x, y over Rational Field
sage: I.groebner_basis()
[y^2, x]
sage: X.dimension()
0
sage: X.base_ring()
Rational Field
sage: X.base_scheme()
Spectrum of Rational Field
sage: X.structure_morphism()
Scheme morphism:
From: Closed subscheme of Affine Space of dimension 2 over Rational Field defined by:
x,
y^2,
x*y^2
To: Spectrum of Rational Field
Defn: Structure map
sage: X.dimension()
0
"""
from sage.schemes.affine.affine_subscheme import (
AlgebraicScheme_subscheme_affine,
AlgebraicScheme_subscheme_affine_field)
if self.base_ring().is_field():
return AlgebraicScheme_subscheme_affine_field(self, X, **kwds)
return AlgebraicScheme_subscheme_affine(self, X, **kwds)
def __init__(self, poly, ambient=None):
"""
Return the affine hypersurface in the space ambient
defined by the polynomial poly.
If ambient is not given, it will be constructed based on
poly.
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
TESTS::
sage: H = AffineHypersurface(x*y-z^3)
sage: H == loads(dumps(H))
True
"""
if not is_MPolynomial(poly):
raise TypeError(
"Defining polynomial (= %s) must be a multivariate polynomial"
% poly)
if ambient is None:
R = poly.parent()
from sage.schemes.affine.affine_space import AffineSpace
ambient = AffineSpace(R.base_ring(), R.ngens())
ambient._coordinate_ring = R
AlgebraicScheme_subscheme_affine.__init__(self, ambient, [poly])