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class DrinfeldModuleAction(Action):
'\n This class implements the module action induced by a Drinfeld\n `\\mathbb{F}_q[T]`-module.\n\n Let `\\phi` be a Drinfeld `\\mathbb{F}_q[T]`-module over a field `K`\n and let `L/K` be a field extension. Let `x \\in L` and let `a` be a\n function ring element; ... |
class DrinfeldModule_charzero(DrinfeldModule):
"\n This class implements Drinfeld `\\mathbb{F}_q[T]`-modules defined\n over fields of `\\mathbb{F}_q[T]`-characteristic zero.\n\n Recall that the `\\mathbb{F}_q[T]`-*characteristic* is defined as the\n kernel of the underlying structure morphism. For gen... |
class DrinfeldModule(Parent, UniqueRepresentation):
"\n This class implements Drinfeld `\\mathbb{F}_q[T]`-modules.\n\n Let `\\mathbb{F}_q[T]` be a polynomial ring with coefficients in a\n finite field `\\mathbb{F}_q` and let `K` be a field. Fix a ring\n morphism `\\gamma: \\mathbb{F}_q[T] \\to K`; we ... |
class DrinfeldModule_finite(DrinfeldModule):
'\n This class implements finite Drinfeld `\\mathbb{F}_q[T]`-modules.\n\n A *finite Drinfeld module* is a Drinfeld module whose base field is\n finite. In this case, the function field characteristic is a prime\n ideal.\n\n For general definitions and he... |
class DrinfeldModuleMorphismAction(Action):
'\n Action of the function ring on the homset of a Drinfeld module.\n\n EXAMPLES::\n\n sage: Fq = GF(5)\n sage: A.<T> = Fq[]\n sage: K.<z> = Fq.extension(3)\n sage: phi = DrinfeldModule(A, [z, 1, z])\n sage: psi = DrinfeldModule(... |
class DrinfeldModuleHomset(Homset):
'\n This class implements the set of morphisms between two Drinfeld\n `\\mathbb{F}_q[T]`-modules.\n\n INPUT:\n\n - ``X`` -- the domain\n\n - ``Y`` -- the codomain\n\n EXAMPLES::\n\n sage: Fq = GF(27)\n sage: A.<T> = Fq[]\n sage: K.<z6> = F... |
class DrinfeldModuleMorphism(Morphism, UniqueRepresentation, metaclass=InheritComparisonClasscallMetaclass):
'\n This class represents Drinfeld `\\mathbb{F}_q[T]`-module morphisms.\n\n Let `\\phi` and `\\psi` be two Drinfeld `\\mathbb{F}_q[T]`-modules over\n a field `K`. A *morphism of Drinfeld modules* ... |
class FunctionFieldExtension(RingExtension_generic):
'\n Abstract base class of function field extensions.\n '
pass
|
class ConstantFieldExtension(FunctionFieldExtension):
'\n Constant field extension.\n\n INPUT:\n\n - ``F`` -- a function field whose constant field is `k`\n\n - ``k_ext`` -- an extension of `k`\n\n '
def __init__(self, F, k_ext):
"\n Initialize.\n\n TESTS::\n\n ... |
def is_FunctionField(x):
"\n Return ``True`` if ``x`` is a function field.\n\n EXAMPLES::\n\n sage: from sage.rings.function_field.function_field import is_FunctionField\n sage: is_FunctionField(QQ)\n False\n sage: is_FunctionField(FunctionField(QQ, 't'))\n True\n "
... |
class FunctionField(Field):
'\n Abstract base class for all function fields.\n\n INPUT:\n\n - ``base_field`` -- field; the base of this function field\n\n - ``names`` -- string that gives the name of the generator\n\n EXAMPLES::\n\n sage: K.<x> = FunctionField(QQ)\n sage: K\n R... |
class FunctionField_polymod(FunctionField):
'\n Function fields defined by a univariate polynomial, as an extension of the\n base field.\n\n INPUT:\n\n - ``polynomial`` -- univariate polynomial over a function field\n\n - ``names`` -- tuple of length 1 or string; variable names\n\n - ``category`... |
class FunctionField_simple(FunctionField_polymod):
'\n Function fields defined by irreducible and separable polynomials\n over rational function fields.\n '
@cached_method
def _inversion_isomorphism(self):
'\n Return an inverted function field isomorphic to ``self`` and isomorphis... |
class FunctionField_char_zero(FunctionField_simple):
'\n Function fields of characteristic zero.\n\n EXAMPLES::\n\n sage: K.<x> = FunctionField(QQ); _.<Y> = K[]\n sage: L.<y> = K.extension(Y^3 - (x^3 - 1)/(x^3 - 2))\n sage: L\n Function field in y defined by y^3 + (-x^3 + 1)/(x^3... |
class FunctionField_global(FunctionField_simple):
'\n Global function fields.\n\n INPUT:\n\n - ``polynomial`` -- monic irreducible and separable polynomial\n\n - ``names`` -- name of the generator of the function field\n\n EXAMPLES::\n\n sage: K.<x> = FunctionField(GF(5)); _.<Y> = K[] ... |
@handle_AA_and_QQbar
def _singular_normal(ideal):
"\n Compute the normalization of the affine algebra defined by ``ideal`` using\n Singular.\n\n The affine algebra is the quotient algebra of a multivariate polynomial\n ring `R` by the ideal. The normalization is by definition the integral\n closure... |
class FunctionField_integral(FunctionField_simple):
'\n Integral function fields.\n\n A function field is integral if it is defined by an irreducible separable\n polynomial, which is integral over the maximal order of the base rational\n function field.\n '
def _maximal_order_basis(self):
... |
class FunctionField_char_zero_integral(FunctionField_char_zero, FunctionField_integral):
'\n Function fields of characteristic zero, defined by an irreducible and\n separable polynomial, integral over the maximal order of the base rational\n function field with a finite constant field.\n '
pass
|
class FunctionField_global_integral(FunctionField_global, FunctionField_integral):
'\n Global function fields, defined by an irreducible and separable polynomial,\n integral over the maximal order of the base rational function field with a\n finite constant field.\n '
pass
|
class RationalFunctionField(FunctionField):
"\n Rational function field in one variable, over an arbitrary base field.\n\n INPUT:\n\n - ``constant_field`` -- arbitrary field\n\n - ``names`` -- string or tuple of length 1\n\n EXAMPLES::\n\n sage: K.<t> = FunctionField(GF(3)); K\n Ratio... |
class RationalFunctionField_char_zero(RationalFunctionField):
'\n Rational function fields of characteristic zero.\n '
@cached_method
def higher_derivation(self):
'\n Return the higher derivation for the function field.\n\n This is also called the Hasse-Schmidt derivation.\n\n... |
class RationalFunctionField_global(RationalFunctionField):
'\n Rational function field over finite fields.\n '
_differentials_space = LazyImport('sage.rings.function_field.differential', 'DifferentialsSpace_global')
def places(self, degree=1):
'\n Return all places of the degree.\n\n... |
class FunctionFieldIdeal(Element):
'\n Base class of fractional ideals of function fields.\n\n INPUT:\n\n - ``ring`` -- ring of the ideal\n\n EXAMPLES::\n\n sage: K.<x> = FunctionField(GF(7))\n sage: O = K.equation_order()\n sage: O.ideal(x^3 + 1)\n Ideal (x^3 + 1) of Maxim... |
class FunctionFieldIdeal_module(FunctionFieldIdeal, Ideal_generic):
'\n A fractional ideal specified by a finitely generated module over\n the integers of the base field.\n\n INPUT:\n\n - ``ring`` -- an order in a function field\n\n - ``module`` -- a module of the order\n\n EXAMPLES:\n\n An i... |
class FunctionFieldIdealInfinite(FunctionFieldIdeal):
'\n Base class of ideals of maximal infinite orders\n '
pass
|
class FunctionFieldIdealInfinite_module(FunctionFieldIdealInfinite, Ideal_generic):
'\n A fractional ideal specified by a finitely generated module over\n the integers of the base field.\n\n INPUT:\n\n - ``ring`` -- order in a function field\n\n - ``module`` -- module\n\n EXAMPLES::\n\n s... |
class IdealMonoid(UniqueRepresentation, Parent):
'\n The monoid of ideals in orders of function fields.\n\n INPUT:\n\n - ``R`` -- order\n\n EXAMPLES::\n\n sage: K.<x> = FunctionField(GF(2))\n sage: O = K.maximal_order()\n sage: M = O.ideal_monoid(); M\n Monoid of ideals of ... |
class FunctionFieldIdeal_polymod(FunctionFieldIdeal):
'\n Fractional ideals of algebraic function fields\n\n INPUT:\n\n - ``ring`` -- order in a function field\n\n - ``hnf`` -- matrix in hermite normal form\n\n - ``denominator`` -- denominator\n\n The rows of ``hnf`` is a basis of the ideal, whi... |
class FunctionFieldIdeal_global(FunctionFieldIdeal_polymod):
'\n Fractional ideals of canonical function fields\n\n INPUT:\n\n - ``ring`` -- order in a function field\n\n - ``hnf`` -- matrix in hermite normal form\n\n - ``denominator`` -- denominator\n\n The rows of ``hnf`` is a basis of the ide... |
class FunctionFieldIdealInfinite_polymod(FunctionFieldIdealInfinite):
'\n Ideals of the infinite maximal order of an algebraic function field.\n\n INPUT:\n\n - ``ring`` -- infinite maximal order of the function field\n\n - ``ideal`` -- ideal in the inverted function field\n\n EXAMPLES::\n\n ... |
class FunctionFieldIdeal_rational(FunctionFieldIdeal):
'\n Fractional ideals of the maximal order of a rational function field.\n\n INPUT:\n\n - ``ring`` -- the maximal order of the rational function field.\n\n - ``gen`` -- generator of the ideal, an element of the function field.\n\n EXAMPLES::\n\... |
class FunctionFieldIdealInfinite_rational(FunctionFieldIdealInfinite):
'\n Fractional ideal of the maximal order of rational function field.\n\n INPUT:\n\n - ``ring`` -- infinite maximal order\n\n - ``gen``-- generator\n\n Note that the infinite maximal order is a principal ideal domain.\n\n EXA... |
class FunctionFieldVectorSpaceIsomorphism(Morphism):
'\n Base class for isomorphisms between function fields and vector spaces.\n\n EXAMPLES::\n\n sage: # needs sage.modules sage.rings.function_field\n sage: K.<x> = FunctionField(QQ); R.<y> = K[]\n sage: L.<y> = K.extension(y^2 - x*y + ... |
class MapVectorSpaceToFunctionField(FunctionFieldVectorSpaceIsomorphism):
'\n Isomorphism from a vector space to a function field.\n\n EXAMPLES::\n\n sage: # needs sage.modules sage.rings.function_field\n sage: K.<x> = FunctionField(QQ); R.<y> = K[]\n sage: L.<y> = K.extension(y^2 - x*y... |
class MapFunctionFieldToVectorSpace(FunctionFieldVectorSpaceIsomorphism):
'\n Isomorphism from a function field to a vector space.\n\n EXAMPLES::\n\n sage: # needs sage.modules sage.rings.function_field\n sage: K.<x> = FunctionField(QQ); R.<y> = K[]\n sage: L.<y> = K.extension(y^2 - x*y... |
class FunctionFieldMorphism(RingHomomorphism):
'\n Base class for morphisms between function fields.\n\n EXAMPLES::\n\n sage: K.<x> = FunctionField(QQ)\n sage: f = K.hom(1/x); f\n Function Field endomorphism of Rational function field in x over Rational Field\n Defn: x |--> 1/x... |
class FunctionFieldMorphism_polymod(FunctionFieldMorphism):
"\n Morphism from a finite extension of a function field to a function field.\n\n EXAMPLES::\n\n sage: # needs sage.rings.finite_rings sage.rings.function_field\n sage: K.<x> = FunctionField(GF(7)); R.<y> = K[]\n sage: L.<y> = ... |
class FunctionFieldMorphism_rational(FunctionFieldMorphism):
'\n Morphism from a rational function field to a function field.\n '
def __init__(self, parent, im_gen, base_morphism):
'\n Initialize.\n\n EXAMPLES::\n\n sage: K.<x> = FunctionField(GF(7))\n sage: ... |
class FunctionFieldConversionToConstantBaseField(Map):
'\n Conversion map from the function field to its constant base field.\n\n EXAMPLES::\n\n sage: K.<x> = FunctionField(QQ)\n sage: QQ.convert_map_from(K)\n Conversion map:\n From: Rational function field in x over Rational F... |
class FunctionFieldToFractionField(FunctionFieldVectorSpaceIsomorphism):
"\n Isomorphism from rational function field to the isomorphic fraction\n field of a polynomial ring.\n\n EXAMPLES::\n\n sage: K = QQ['x'].fraction_field()\n sage: L = K.function_field()\n sage: f = K.coerce_map... |
class FractionFieldToFunctionField(FunctionFieldVectorSpaceIsomorphism):
"\n Isomorphism from a fraction field of a polynomial ring to the isomorphic\n function field.\n\n EXAMPLES::\n\n sage: K = QQ['x'].fraction_field()\n sage: L = K.function_field()\n sage: f = L.coerce_map_from(K... |
class FunctionFieldCompletion(Map):
"\n Completions on function fields.\n\n INPUT:\n\n - ``field`` -- function field\n\n - ``place`` -- place of the function field\n\n - ``name`` -- string for the name of the series variable\n\n - ``prec`` -- positive integer; default precision\n\n - ``gen_na... |
class FunctionFieldRingMorphism(SetMorphism):
'\n Ring homomorphism.\n '
def _repr_(self) -> str:
'\n Return the string representation of the map.\n\n EXAMPLES::\n\n sage: # needs sage.rings.finite_rings sage.rings.function_field\n sage: K.<x> = FunctionField... |
class FunctionFieldLinearMap(SetMorphism):
'\n Linear map to function fields.\n '
def _repr_(self) -> str:
'\n Return the string representation of the map.\n\n EXAMPLES::\n\n sage: # needs sage.rings.finite_rings sage.rings.function_field\n sage: K.<x> = Func... |
class FunctionFieldLinearMapSection(SetMorphism):
'\n Section of linear map from function fields.\n '
def _repr_(self) -> str:
'\n Return the string representation of the map.\n\n EXAMPLES::\n\n sage: # needs sage.rings.finite_rings sage.rings.function_field\n ... |
class FunctionFieldOrder_base(CachedRepresentation, Parent):
"\n Base class for orders in function fields.\n\n INPUT:\n\n - ``field`` -- function field\n\n EXAMPLES::\n\n sage: F = FunctionField(QQ,'y')\n sage: F.maximal_order()\n Maximal order of Rational function field in y over... |
class FunctionFieldOrder(FunctionFieldOrder_base):
'\n Base class for orders in function fields.\n '
def _repr_(self):
"\n Return the string representation.\n\n EXAMPLES::\n\n sage: FunctionField(QQ,'y').maximal_order()\n Maximal order of Rational function fi... |
class FunctionFieldOrderInfinite(FunctionFieldOrder_base):
'\n Base class for infinite orders in function fields.\n '
def _repr_(self):
"\n EXAMPLES::\n\n sage: FunctionField(QQ,'y').maximal_order_infinite()\n Maximal infinite order of Rational function field in y o... |
class FunctionFieldMaximalOrder(UniqueRepresentation, FunctionFieldOrder):
'\n Base class of maximal orders of function fields.\n '
def _repr_(self):
"\n Return the string representation of the order.\n\n EXAMPLES::\n\n sage: FunctionField(QQ,'y').maximal_order()._repr_... |
class FunctionFieldMaximalOrderInfinite(FunctionFieldMaximalOrder, FunctionFieldOrderInfinite):
'\n Base class of maximal infinite orders of function fields.\n '
def _repr_(self):
"\n EXAMPLES::\n\n sage: FunctionField(QQ,'y').maximal_order_infinite()\n Maximal infi... |
class FunctionFieldOrder_basis(FunctionFieldOrder):
'\n Order given by a basis over the maximal order of the base field.\n\n INPUT:\n\n - ``basis`` -- list of elements of the function field\n\n - ``check`` -- (default: ``True``) if ``True``, check whether the module\n that ``basis`` generates for... |
class FunctionFieldOrderInfinite_basis(FunctionFieldOrderInfinite):
'\n Order given by a basis over the infinite maximal order of the base field.\n\n INPUT:\n\n - ``basis`` -- elements of the function field\n\n - ``check`` -- boolean (default: ``True``); if ``True``, check the basis generates\n a... |
class FunctionFieldMaximalOrder_polymod(FunctionFieldMaximalOrder):
'\n Maximal orders of extensions of function fields.\n '
def __init__(self, field, ideal_class=FunctionFieldIdeal_polymod):
'\n Initialize.\n\n TESTS::\n\n sage: # needs sage.rings.finite_rings\n ... |
class FunctionFieldMaximalOrderInfinite_polymod(FunctionFieldMaximalOrderInfinite):
'\n Maximal infinite orders of function fields.\n\n INPUT:\n\n - ``field`` -- function field\n\n EXAMPLES::\n\n sage: K.<x> = FunctionField(GF(2)); _.<t> = PolynomialRing(K) # needs sage.rings.... |
class FunctionFieldMaximalOrder_global(FunctionFieldMaximalOrder_polymod):
'\n Maximal orders of global function fields.\n\n INPUT:\n\n - ``field`` -- function field to which this maximal order belongs\n\n EXAMPLES::\n\n sage: K.<x> = FunctionField(GF(7)); R.<y> = K[] ... |
class FunctionFieldMaximalOrder_rational(FunctionFieldMaximalOrder):
'\n Maximal orders of rational function fields.\n\n INPUT:\n\n - ``field`` -- a function field\n\n EXAMPLES::\n\n sage: K.<t> = FunctionField(GF(19)); K\n Rational function field in t over Finite Field of size 19\n ... |
class FunctionFieldMaximalOrderInfinite_rational(FunctionFieldMaximalOrderInfinite):
'\n Maximal infinite orders of rational function fields.\n\n INPUT:\n\n - ``field`` -- a rational function field\n\n EXAMPLES::\n\n sage: K.<t> = FunctionField(GF(19)); K\n Rational function field in t o... |
class FunctionFieldPlace(Element):
'\n Places of function fields.\n\n INPUT:\n\n - ``parent`` -- place set of a function field\n\n - ``prime`` -- prime ideal associated with the place\n\n EXAMPLES::\n\n sage: K.<x> = FunctionField(GF(2)); _.<Y> = K[]\n sage: L.<y> = K.extension(Y^3 + ... |
class PlaceSet(UniqueRepresentation, Parent):
'\n Sets of Places of function fields.\n\n INPUT:\n\n - ``field`` -- function field\n\n EXAMPLES::\n\n sage: K.<x> = FunctionField(GF(2)); _.<Y> = K[]\n sage: L.<y> = K.extension(Y^3 + x^3*Y + x) # needs s... |
class FunctionFieldPlace_polymod(FunctionFieldPlace):
'\n Places of extensions of function fields.\n '
def place_below(self):
'\n Return the place lying below the place.\n\n EXAMPLES::\n\n sage: # needs sage.rings.finite_rings\n sage: K.<x> = FunctionField(GF... |
class FunctionFieldPlace_rational(FunctionFieldPlace):
'\n Places of rational function fields.\n '
def degree(self):
'\n Return the degree of the place.\n\n EXAMPLES::\n\n sage: F.<x> = FunctionField(GF(2))\n sage: O = F.maximal_order()\n sage: i =... |
class FunctionFieldValuationFactory(UniqueFactory):
'\n Create a valuation on ``domain`` corresponding to ``prime``.\n\n INPUT:\n\n - ``domain`` -- a function field\n\n - ``prime`` -- a place of the function field, a valuation on a subring, or\n a valuation on another function field together with... |
class FunctionFieldValuation_base(DiscretePseudoValuation):
'\n Abstract base class for any discrete (pseudo-)valuation on a function\n field.\n\n TESTS::\n\n sage: K.<x> = FunctionField(QQ)\n sage: v = K.valuation(x) # indirect doctest\n sage: from sage.rings.function_field.valuati... |
class DiscreteFunctionFieldValuation_base(DiscreteValuation):
'\n Base class for discrete valuations on function fields.\n\n TESTS::\n\n sage: K.<x> = FunctionField(QQ)\n sage: v = K.valuation(x) # indirect doctest\n sage: from sage.rings.function_field.valuation import DiscreteFunctio... |
class RationalFunctionFieldValuation_base(FunctionFieldValuation_base):
'\n Base class for valuations on rational function fields.\n\n TESTS::\n\n sage: K.<x> = FunctionField(GF(2))\n sage: v = K.valuation(x) # indirect doctest\n sage: from sage.rings.function_field.valuation import Ra... |
class ClassicalFunctionFieldValuation_base(DiscreteFunctionFieldValuation_base):
'\n Base class for discrete valuations on rational function fields that come\n from points on the projective line.\n\n TESTS::\n\n sage: K.<x> = FunctionField(GF(5))\n sage: v = K.valuation(x) # indirect docte... |
class InducedRationalFunctionFieldValuation_base(FunctionFieldValuation_base):
'\n Base class for function field valuation induced by a valuation on the\n underlying polynomial ring.\n\n TESTS::\n\n sage: K.<x> = FunctionField(QQ)\n sage: v = K.valuation(x^2 + 1) # indirect doctest\n\n '... |
class FiniteRationalFunctionFieldValuation(InducedRationalFunctionFieldValuation_base, ClassicalFunctionFieldValuation_base, RationalFunctionFieldValuation_base):
'\n Valuation of a finite place of a function field.\n\n EXAMPLES::\n\n sage: K.<x> = FunctionField(QQ)\n sage: v = K.valuation(x +... |
class NonClassicalRationalFunctionFieldValuation(InducedRationalFunctionFieldValuation_base, RationalFunctionFieldValuation_base):
"\n Valuation induced by a valuation on the underlying polynomial ring which is\n non-classical.\n\n EXAMPLES::\n\n sage: K.<x> = FunctionField(QQ)\n sage: v = ... |
class FunctionFieldFromLimitValuation(FiniteExtensionFromLimitValuation, DiscreteFunctionFieldValuation_base):
'\n A valuation on a finite extensions of function fields `L=K[y]/(G)` where `K` is\n another function field.\n\n EXAMPLES::\n\n sage: K.<x> = FunctionField(QQ)\n sage: R.<y> = K[]... |
class FunctionFieldMappedValuation_base(FunctionFieldValuation_base, MappedValuation_base):
'\n A valuation on a function field which relies on a ``base_valuation`` on an\n isomorphic function field.\n\n EXAMPLES::\n\n sage: K.<x> = FunctionField(GF(2))\n sage: v = K.valuation(1/x); v\n ... |
class FunctionFieldMappedValuationRelative_base(FunctionFieldMappedValuation_base):
'\n A valuation on a function field which relies on a ``base_valuation`` on an\n isomorphic function field and which is such that the map from and to the\n other function field is the identity on the constant field.\n\n ... |
class RationalFunctionFieldMappedValuation(FunctionFieldMappedValuationRelative_base, RationalFunctionFieldValuation_base):
'\n Valuation on a rational function field that is implemented after a map to\n an isomorphic rational function field.\n\n EXAMPLES::\n\n sage: K.<x> = FunctionField(QQ)\n ... |
class InfiniteRationalFunctionFieldValuation(FunctionFieldMappedValuationRelative_base, RationalFunctionFieldValuation_base, ClassicalFunctionFieldValuation_base):
'\n Valuation of the infinite place of a function field.\n\n EXAMPLES::\n\n sage: K.<x> = FunctionField(QQ)\n sage: v = K.valuatio... |
class FunctionFieldExtensionMappedValuation(FunctionFieldMappedValuationRelative_base):
'\n A valuation on a finite extensions of function fields `L=K[y]/(G)` where `K` is\n another function field which redirects to another ``base_valuation`` on an\n isomorphism function field `M=K[y]/(H)`.\n\n The is... |
class FunctionFieldValuationRing(UniqueRepresentation, Parent):
'\n Base class for valuation rings of function fields.\n\n INPUT:\n\n - ``field`` -- function field\n\n - ``place`` -- place of the function field\n\n EXAMPLES::\n\n sage: K.<x> = FunctionField(GF(2)); _.<Y> = K[]\n sage:... |
class ProductTree():
'\n A simple binary product tree, i.e., a tree of ring elements in\n which every node equals the product of its children.\n (In particular, the *root* equals the product of all *leaves*.)\n\n Product trees are a very useful building block for fast computer\n algebra. For exampl... |
def prod_with_derivative(pairs):
'\n Given an iterable of pairs `(f, \\partial f)` of ring elements,\n return the pair `(\\prod f, \\partial \\prod f)`, assuming `\\partial`\n is an operator obeying the standard product rule.\n\n This function is entirely algebraic, hence still works when the\n ele... |
def is_RingHomset(H):
'\n Return ``True`` if ``H`` is a space of homomorphisms between two rings.\n\n EXAMPLES::\n\n sage: from sage.rings.homset import is_RingHomset as is_RH\n sage: is_RH(Hom(ZZ, QQ))\n True\n sage: is_RH(ZZ)\n False\n sage: is_RH(Hom(RR, CC)) ... |
def RingHomset(R, S, category=None):
'\n Construct a space of homomorphisms between the rings ``R`` and ``S``.\n\n For more on homsets, see :func:`Hom()`.\n\n EXAMPLES::\n\n sage: Hom(ZZ, QQ) # indirect doctest\n Set of Homomorphisms from Integer Ring to Rational Field\n\n '
if quoti... |
class RingHomset_generic(HomsetWithBase):
'\n A generic space of homomorphisms between two rings.\n\n EXAMPLES::\n\n sage: Hom(ZZ, QQ)\n Set of Homomorphisms from Integer Ring to Rational Field\n sage: QQ.Hom(ZZ)\n Set of Homomorphisms from Rational Field to Integer Ring\n '
... |
class RingHomset_quo_ring(RingHomset_generic):
'\n Space of ring homomorphisms where the domain is a (formal) quotient\n ring.\n\n EXAMPLES::\n\n sage: R.<x,y> = PolynomialRing(QQ, 2)\n sage: S.<a,b> = R.quotient(x^2 + y^2) # needs sage.libs.singula... |
def Ideal(*args, **kwds):
"\n Create the ideal in ring with given generators.\n\n There are some shorthand notations for creating an ideal, in\n addition to using the :func:`Ideal` function:\n\n - ``R.ideal(gens, coerce=True)``\n - ``gens*R``\n - ``R*gens``\n\n INPUT:\n\n - ``R`` - A r... |
def is_Ideal(x):
"\n Return ``True`` if object is an ideal of a ring.\n\n EXAMPLES:\n\n A simple example involving the ring of integers. Note\n that Sage does not interpret rings objects themselves as ideals.\n However, one can still explicitly construct these ideals::\n\n sage: from sage.ri... |
class Ideal_generic(MonoidElement):
'\n An ideal.\n\n See :func:`Ideal()`.\n '
def __init__(self, ring, gens, coerce=True):
'\n Initialize this ideal.\n\n INPUT:\n\n - ``ring`` -- A ring\n\n - ``gens`` -- The generators for this ideal\n\n - ``coerce`` -- (d... |
class Ideal_principal(Ideal_generic):
'\n A principal ideal.\n\n See :func:`Ideal()`.\n '
def __repr__(self):
'\n Return a string representation of ``self``.\n\n EXAMPLES::\n\n sage: R.<x> = ZZ[]\n sage: I = R.ideal(x)\n sage: I # indirect docte... |
class Ideal_pid(Ideal_principal):
'\n An ideal of a principal ideal domain.\n\n See :func:`Ideal()`.\n '
def __init__(self, ring, gen):
'\n Initialize ``self``.\n\n EXAMPLES::\n\n sage: I = 8*ZZ\n sage: I\n Principal ideal (8) of Integer Ring\n ... |
class Ideal_fractional(Ideal_generic):
'\n Fractional ideal of a ring.\n\n See :func:`Ideal()`.\n '
def __repr__(self):
"\n Return a string representation of ``self``.\n\n EXAMPLES::\n\n sage: from sage.rings.ideal import Ideal_fractional\n sage: x = polyg... |
def Cyclic(R, n=None, homog=False, singular=None):
"\n Ideal of cyclic ``n``-roots from 1-st ``n`` variables of ``R`` if ``R`` is\n coercible to :class:`Singular <sage.interfaces.singular.Singular>`.\n\n INPUT:\n\n - ``R`` -- base ring to construct ideal for\n\n - ``n`` -- number of cyclic roots ... |
def Katsura(R, n=None, homog=False, singular=None):
'\n ``n``-th katsura ideal of ``R`` if ``R`` is coercible to\n :class:`Singular <sage.interfaces.singular.Singular>`.\n\n INPUT:\n\n - ``R`` -- base ring to construct ideal for\n\n - ``n`` -- (default: ``None``) which katsura ideal of ``R``. If ``... |
def FieldIdeal(R):
"\n Let ``q = R.base_ring().order()`` and `(x_0,...,x_n)` ``= R.gens()`` then\n if `q` is finite this constructor returns\n\n .. MATH::\n\n \\langle x_0^q - x_0, ... , x_n^q - x_n \\rangle.\n\n We call this ideal the field ideal and the generators the field\n equations.\n\... |
def IdealMonoid(R):
"\n Return the monoid of ideals in the ring ``R``.\n\n EXAMPLES::\n\n sage: R = QQ['x']\n sage: from sage.rings.ideal_monoid import IdealMonoid\n sage: IdealMonoid(R)\n Monoid of ideals of Univariate Polynomial Ring in x over Rational Field\n "
return I... |
class IdealMonoid_c(Parent):
'\n The monoid of ideals in a commutative ring.\n\n TESTS::\n\n sage: R = QQ[\'x\']\n sage: from sage.rings.ideal_monoid import IdealMonoid\n sage: M = IdealMonoid(R)\n sage: TestSuite(M).run()\n Failure in _test_category:\n ...\n ... |
class _uniq():
def __new__(cls, *args):
'\n This ensures uniqueness of these objects.\n\n EXAMPLES::\n\n sage: sage.rings.infinity.UnsignedInfinityRing_class() is sage.rings.infinity.UnsignedInfinityRing_class()\n True\n '
if (cls in _obj):
r... |
class AnInfinity():
'\n TESTS::\n\n sage: oo == oo\n True\n sage: oo < oo\n False\n sage: -oo < oo\n True\n sage: -oo < 3 < oo\n True\n\n sage: unsigned_infinity == 3\n False\n sage: unsigned_infinity == unsigned_infinity\n Tru... |
class UnsignedInfinityRing_class(Singleton, Ring):
def __init__(self):
"\n Initialize ``self``.\n\n TESTS::\n\n sage: sage.rings.infinity.UnsignedInfinityRing_class() is sage.rings.infinity.UnsignedInfinityRing_class() is UnsignedInfinityRing\n True\n\n Sage can... |
class LessThanInfinity(_uniq, RingElement):
def __init__(self, parent=UnsignedInfinityRing):
'\n Initialize ``self``.\n\n EXAMPLES::\n\n sage: sage.rings.infinity.LessThanInfinity() is UnsignedInfinityRing(5)\n True\n '
RingElement.__init__(self, parent)... |
class UnsignedInfinity(_uniq, AnInfinity, InfinityElement):
_sign = 0
_sign_char = ''
def __init__(self):
'\n Initialize ``self``.\n\n TESTS::\n\n sage: sage.rings.infinity.UnsignedInfinity() is sage.rings.infinity.UnsignedInfinity() is unsigned_infinity\n True... |
def is_Infinite(x) -> bool:
'\n This is a type check for infinity elements.\n\n EXAMPLES::\n\n sage: sage.rings.infinity.is_Infinite(oo)\n True\n sage: sage.rings.infinity.is_Infinite(-oo)\n True\n sage: sage.rings.infinity.is_Infinite(unsigned_infinity)\n True\n ... |
class SignError(ArithmeticError):
'\n Sign error exception.\n '
pass
|
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