def __init__(self, polynomial_ring, category):
r"""
Create the ring of CFiniteSequences over ``base_ring``
INPUT:
- ``base_ring`` -- the base ring for the o.g.f (either ``QQ`` or ``ZZ``)
- ``names`` -- an iterable of variables (shuould contain only one variable)
- ``category`` -- the category of the ring (default: ``Fields()``)
TESTS::
sage: C.<y> = CFiniteSequences(QQ); C
The ring of C-Finite sequences in y over Rational Field
sage: C.<x> = CFiniteSequences(QQ); C
The ring of C-Finite sequences in x over Rational Field
sage: C.<x> = CFiniteSequences(ZZ); C
The ring of C-Finite sequences in x over Integer Ring
sage: C.<x,y> = CFiniteSequences(ZZ)
Traceback (most recent call last):
...
NotImplementedError: Multidimensional o.g.f. not implemented.
sage: C.<x> = CFiniteSequences(CC)
Traceback (most recent call last):
...
ValueError: O.g.f. base not rational.
"""
base_ring = polynomial_ring.base_ring()
self._polynomial_ring = polynomial_ring
self._fraction_field = FractionField(self._polynomial_ring)
CommutativeRing.__init__(self, base_ring, self._polynomial_ring.gens(), category)
def __init__(self, polynomial_ring, category):
r"""
Create the ring of CFiniteSequences over ``base_ring``
INPUT:
- ``base_ring`` -- the base ring for the o.g.f (either ``QQ`` or ``ZZ``)
- ``names`` -- an iterable of variables (shuould contain only one variable)
- ``category`` -- the category of the ring (default: ``Fields()``)
TESTS::
sage: C.<y> = CFiniteSequences(QQ); C
The ring of C-Finite sequences in y over Rational Field
sage: C.<x> = CFiniteSequences(QQ); C
The ring of C-Finite sequences in x over Rational Field
sage: C.<x> = CFiniteSequences(ZZ); C
The ring of C-Finite sequences in x over Integer Ring
sage: C.<x,y> = CFiniteSequences(ZZ)
Traceback (most recent call last):
...
NotImplementedError: Multidimensional o.g.f. not implemented.
sage: C.<x> = CFiniteSequences(CC)
Traceback (most recent call last):
...
ValueError: O.g.f. base not rational.
"""
base_ring = polynomial_ring.base_ring()
self._polynomial_ring = polynomial_ring
self._fraction_field = FractionField(self._polynomial_ring)
CommutativeRing.__init__(self, base_ring, self._polynomial_ring.gens(),
category)
def __init__(self,
base_ring,
num_gens,
name_list,
order='negdeglex',
default_prec=10,
sparse=False):
"""
Initializes a multivariate power series ring. See PowerSeriesRing
for complete documentation.
INPUT
- ``base_ring`` - a commutative ring
- ``num_gens`` - number of generators
- ``name_list`` - List of indeterminate names or a single name.
If a single name is given, indeterminates will be this name
followed by a number from 0 to num_gens - 1. If a list is
given, these will be the indeterminate names and the length
of the list must be equal to num_gens.
- ``order`` - ordering of variables; default is
negative degree lexicographic
- ``default_prec`` - The default total-degree precision for
elements. The default value of default_prec is 10.
- ``sparse`` - whether or not power series are sparse
EXAMPLES::
sage: R.<t,u,v> = PowerSeriesRing(QQ)
sage: g = 1 + v + 3*u*t^2 - 2*v^2*t^2
sage: g = g.add_bigoh(5); g
1 + v + 3*t^2*u - 2*t^2*v^2 + O(t, u, v)^5
sage: g in R
True
TESTS:
By :trac:`14084`, the multi-variate power series ring belongs to the
category of integral domains, if the base ring does::
sage: P = ZZ[['x','y']]
sage: P.category()
Category of integral domains
sage: TestSuite(P).run()
Otherwise, it belongs to the category of commutative rings::
sage: P = Integers(15)[['x','y']]
sage: P.category()
Category of commutative rings
sage: TestSuite(P).run()
"""
order = TermOrder(order, num_gens)
self._term_order = order
if not base_ring.is_commutative():
raise TypeError("Base ring must be a commutative ring.")
n = int(num_gens)
if n < 0:
raise ValueError(
"Multivariate Polynomial Rings must have more than 0 variables."
)
self._ngens = n
self._has_singular = False #cannot convert to Singular by default
# Multivariate power series rings inherit from power series rings. But
# apparently we can not call their initialisation. Instead, initialise
# CommutativeRing and Nonexact:
CommutativeRing.__init__(self,
base_ring,
name_list,
category=_IntegralDomains if base_ring
in _IntegralDomains else _CommutativeRings)
Nonexact.__init__(self, default_prec)
# underlying polynomial ring in which to represent elements
self._poly_ring_ = PolynomialRing(base_ring,
self.variable_names(),
sparse=sparse,
order=order)
# because sometimes PowerSeriesRing_generic calls self.__poly_ring
self._PowerSeriesRing_generic__poly_ring = self._poly_ring()
# background univariate power series ring
self._bg_power_series_ring = PowerSeriesRing(self._poly_ring_,
'Tbg',
sparse=sparse,
default_prec=default_prec)
self._bg_indeterminate = self._bg_power_series_ring.gen()
self._is_sparse = sparse
self._params = (base_ring, num_gens, name_list, order, default_prec,
sparse)
self._populate_coercion_lists_()
def __init__(self, base_ring, num_gens, name_list,
order='negdeglex', default_prec=10, sparse=False):
"""
Initializes a multivariate power series ring. See PowerSeriesRing
for complete documentation.
INPUT
- ``base_ring`` - a commutative ring
- ``num_gens`` - number of generators
- ``name_list`` - List of indeterminate names or a single name.
If a single name is given, indeterminates will be this name
followed by a number from 0 to num_gens - 1. If a list is
given, these will be the indeterminate names and the length
of the list must be equal to num_gens.
- ``order`` - ordering of variables; default is
negative degree lexicographic
- ``default_prec`` - The default total-degree precision for
elements. The default value of default_prec is 10.
- ``sparse`` - whether or not power series are sparse
EXAMPLES::
sage: R.<t,u,v> = PowerSeriesRing(QQ)
sage: g = 1 + v + 3*u*t^2 - 2*v^2*t^2
sage: g = g.add_bigoh(5); g
1 + v + 3*t^2*u - 2*t^2*v^2 + O(t, u, v)^5
sage: g in R
True
TESTS:
By :trac:`14084`, the multi-variate power series ring belongs to the
category of integral domains, if the base ring does::
sage: P = ZZ[['x','y']]
sage: P.category()
Category of integral domains
sage: TestSuite(P).run()
Otherwise, it belongs to the category of commutative rings::
sage: P = Integers(15)[['x','y']]
sage: P.category()
Category of commutative rings
sage: TestSuite(P).run()
"""
order = TermOrder(order,num_gens)
self._term_order = order
if not base_ring.is_commutative():
raise TypeError("Base ring must be a commutative ring.")
n = int(num_gens)
if n < 0:
raise ValueError("Multivariate Polynomial Rings must have more than 0 variables.")
self._ngens = n
self._has_singular = False #cannot convert to Singular by default
# Multivariate power series rings inherit from power series rings. But
# apparently we can not call their initialisation. Instead, initialise
# CommutativeRing and Nonexact:
CommutativeRing.__init__(self, base_ring, name_list, category =
_IntegralDomains if base_ring in
_IntegralDomains else _CommutativeRings)
Nonexact.__init__(self, default_prec)
# underlying polynomial ring in which to represent elements
self._poly_ring_ = PolynomialRing(base_ring, self.variable_names(), sparse=sparse, order=order)
# because sometimes PowerSeriesRing_generic calls self.__poly_ring
self._PowerSeriesRing_generic__poly_ring = self._poly_ring()
# background univariate power series ring
self._bg_power_series_ring = PowerSeriesRing(self._poly_ring_, 'Tbg', sparse=sparse, default_prec=default_prec)
self._bg_indeterminate = self._bg_power_series_ring.gen()
self._is_sparse = sparse
self._params = (base_ring, num_gens, name_list,
order, default_prec, sparse)
self._populate_coercion_lists_()