예제 #1
0
    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)
예제 #2
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    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)
예제 #3
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    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_()
예제 #4
0
    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_()