def Katsura(R, n=None, homog=False, singular=singular_default):
"""
``n``-th katsura ideal of ``R`` if ``R`` is coercible to
:class:`Singular <sage.interfaces.singular.Singular>`.
INPUT:
- ``R`` -- base ring to construct ideal for
- ``n`` -- (default: ``None``) which katsura ideal of ``R``. If ``None``,
then ``n`` is set to ``R.ngens()``.
- ``homog`` -- if ``True`` a homogeneous ideal is returned
using the last variable in the ideal (default: ``False``)
- ``singular`` -- singular instance to use
EXAMPLES::
sage: P.<x,y,z> = PolynomialRing(QQ,3)
sage: I = sage.rings.ideal.Katsura(P,3); I
Ideal (x + 2*y + 2*z - 1, x^2 + 2*y^2 + 2*z^2 - x, 2*x*y + 2*y*z - y)
of Multivariate Polynomial Ring in x, y, z over Rational Field
::
sage: Q.<x> = PolynomialRing(QQ,1)
sage: J = sage.rings.ideal.Katsura(Q,1); J
Ideal (x - 1) of Multivariate Polynomial Ring in x over Rational Field
"""
from rational_field import RationalField
if n:
if n > R.ngens():
raise ArithmeticError("n must be <= R.ngens().")
else:
n = R.ngens()
singular.lib("poly")
R2 = R.change_ring(RationalField())
R2._singular_().set_ring()
if not homog:
I = singular.katsura(n)
else:
I = singular.katsura(n).homog(R2.gen(n - 1))
return R2.ideal(I).change_ring(R)
def Cyclic(R, n=None, homog=False, singular=singular_default):
"""
Ideal of cyclic ``n``-roots from 1-st ``n`` variables of ``R`` if ``R`` is
coercible to :class:`Singular <sage.interfaces.singular.Singular>`.
INPUT:
- ``R`` -- base ring to construct ideal for
- ``n`` -- number of cyclic roots (default: ``None``). If ``None``, then
``n`` is set to ``R.ngens()``.
- ``homog`` -- (default: ``False``) if ``True`` a homogeneous ideal is
returned using the last variable in the ideal
- ``singular`` -- singular instance to use
.. NOTE::
``R`` will be set as the active ring in
:class:`Singular <sage.interfaces.singular.Singular>`
EXAMPLES:
An example from a multivariate polynomial ring over the
rationals::
sage: P.<x,y,z> = PolynomialRing(QQ,3,order='lex')
sage: I = sage.rings.ideal.Cyclic(P)
sage: I
Ideal (x + y + z, x*y + x*z + y*z, x*y*z - 1) of Multivariate Polynomial
Ring in x, y, z over Rational Field
sage: I.groebner_basis()
[x + y + z, y^2 + y*z + z^2, z^3 - 1]
We compute a Groebner basis for cyclic 6, which is a standard
benchmark and test ideal::
sage: R.<x,y,z,t,u,v> = QQ['x,y,z,t,u,v']
sage: I = sage.rings.ideal.Cyclic(R,6)
sage: B = I.groebner_basis()
sage: len(B)
45
"""
from rational_field import RationalField
if n:
if n > R.ngens():
raise ArithmeticError("n must be <= R.ngens()")
else:
n = R.ngens()
singular.lib("poly")
R2 = R.change_ring(RationalField())
R2._singular_().set_ring()
if not homog:
I = singular.cyclic(n)
else:
I = singular.cyclic(n).homog(R2.gen(n - 1))
return R2.ideal(I).change_ring(R)
