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class NonFinalInductiveValuation(FiniteInductiveValuation, DiscreteValuation):
'\n Abstract base class for iterated :mod:`augmented valuations <sage.rings.valuation.augmented_valuation>`\n on top of a :mod:`Gauss valuation <sage.rings.valuation.gauss_valuation>` which can be extended further\n through :m... |
class FinalInductiveValuation(InductiveValuation):
'\n Abstract base class for an inductive valuation which cannot be augmented further.\n\n TESTS::\n\n sage: R.<x> = QQ[]\n sage: v = GaussValuation(R, valuations.TrivialValuation(QQ))\n sage: w = v.augmentation(x^2 + x + 1, infinity)\n ... |
class InfiniteInductiveValuation(FinalInductiveValuation, InfiniteDiscretePseudoValuation):
'\n Abstract base class for an inductive valuation which is not discrete, i.e.,\n which assigns infinite valuation to its last key polynomial.\n\n EXAMPLES::\n\n sage: R.<x> = QQ[]\n sage: v = GaussV... |
def _lift_to_maximal_precision(c):
'\n Lift ``c`` to maximal precision if the parent is not exact.\n\n EXAMPLES::\n\n sage: R = Zp(2,5)\n sage: x = R(1,2); x\n 1 + O(2^2)\n sage: from sage.rings.valuation.inductive_valuation import _lift_to_maximal_precision\n sage: _lift_... |
class LimitValuationFactory(UniqueFactory):
'\n Return a limit valuation which sends the polynomial ``G`` to infinity and\n is greater than or equal than ``base_valuation``.\n\n INPUT:\n\n - ``base_valuation`` -- a discrete (pseudo-)valuation on a polynomial ring\n which is a discrete valuation o... |
class LimitValuation_generic(DiscretePseudoValuation):
'\n Base class for limit valuations.\n\n A limit valuation is realized as an approximation of a valuation and means\n to improve that approximation when necessary.\n\n EXAMPLES::\n\n sage: # needs sage.rings.function_field\n sage: K.... |
class MacLaneLimitValuation(LimitValuation_generic, InfiniteDiscretePseudoValuation):
'\n A limit valuation that is a pseudo-valuation on polynomial ring `K[x]`\n which sends a square-free polynomial `G` to infinity.\n\n This uses the MacLane algorithm to compute the next element in the limit.\n\n It ... |
class MappedValuation_base(DiscretePseudoValuation):
'\n A valuation which is implemented through another proxy "base" valuation.\n\n EXAMPLES::\n\n sage: # needs sage.rings.function_field\n sage: K.<x> = FunctionField(QQ)\n sage: R.<y> = K[]\n sage: L.<y> = K.extension(y^2 - x)\... |
class FiniteExtensionFromInfiniteValuation(MappedValuation_base, DiscreteValuation):
'\n A valuation on a quotient of the form `L=K[x]/(G)` with an irreducible `G`\n which is internally backed by a pseudo-valuations on `K[x]` which sends `G`\n to infinity.\n\n INPUT:\n\n - ``parent`` -- the contain... |
class FiniteExtensionFromLimitValuation(FiniteExtensionFromInfiniteValuation):
'\n An extension of a valuation on a finite field extensions `L=K[x]/(G)` which\n is induced by an infinite limit valuation on `K[x]`.\n\n EXAMPLES::\n\n sage: # needs sage.rings.function_field\n sage: K.<x> = Fu... |
class ScaledValuationFactory(UniqueFactory):
'\n Return a valuation which scales the valuation ``base`` by the factor ``s``.\n\n EXAMPLES::\n\n sage: 3*ZZ.valuation(2) # indirect doctest\n 3 * 2-adic valuation\n '
def create_key(self, base, s):
'\n Create a key which uni... |
class ScaledValuation_generic(DiscreteValuation):
'\n A valuation which scales another ``base_valuation`` by a finite positive factor ``s``.\n\n EXAMPLES::\n\n sage: v = 3*ZZ.valuation(3); v\n 3 * 3-adic valuation\n\n TESTS::\n\n sage: TestSuite(v).run() # long time ... |
class TrivialValuationFactory(UniqueFactory):
'\n Create a trivial valuation on ``domain``.\n\n EXAMPLES::\n\n sage: v = valuations.TrivialValuation(QQ); v\n Trivial valuation on Rational Field\n sage: v(1)\n 0\n '
def __init__(self, clazz, parent, *args, **kwargs):
... |
class TrivialDiscretePseudoValuation_base(DiscretePseudoValuation):
'\n Base class for code shared by trivial valuations.\n\n EXAMPLES::\n\n sage: v = valuations.TrivialPseudoValuation(ZZ); v\n Trivial pseudo-valuation on Integer Ring\n\n TESTS::\n\n sage: TestSuite(v).run() # long t... |
class TrivialDiscretePseudoValuation(TrivialDiscretePseudoValuation_base, InfiniteDiscretePseudoValuation):
'\n The trivial pseudo-valuation that is `\\infty` everywhere.\n\n EXAMPLES::\n\n sage: v = valuations.TrivialPseudoValuation(QQ); v\n Trivial pseudo-valuation on Rational Field\n\n T... |
class TrivialDiscreteValuation(TrivialDiscretePseudoValuation_base, DiscreteValuation):
'\n The trivial valuation that is zero on non-zero elements.\n\n EXAMPLES::\n\n sage: v = valuations.TrivialValuation(QQ); v\n Trivial valuation on Rational Field\n\n TESTS::\n\n sage: TestSuite(v... |
class DiscretePseudoValuation(Morphism):
'\n Abstract base class for discrete pseudo-valuations, i.e., discrete\n valuations which might send more that just zero to infinity.\n\n INPUT:\n\n - ``domain`` -- an integral domain\n\n EXAMPLES::\n\n sage: v = ZZ.valuation(2); v # indirect doctest... |
class InfiniteDiscretePseudoValuation(DiscretePseudoValuation):
'\n Abstract base class for infinite discrete pseudo-valuations, i.e., discrete\n pseudo-valuations which are not discrete valuations.\n\n EXAMPLES::\n\n sage: v = QQ.valuation(2)\n sage: R.<x> = QQ[]\n sage: v = GaussVa... |
class NegativeInfiniteDiscretePseudoValuation(InfiniteDiscretePseudoValuation):
'\n Abstract base class for pseudo-valuations which attain the value `\\infty`\n and `-\\infty`, i.e., whose domain contains an element of valuation `\\infty`\n and its inverse.\n\n EXAMPLES::\n\n sage: R.<x> = QQ[]... |
class DiscreteValuation(DiscretePseudoValuation):
'\n Abstract base class for discrete valuations.\n\n EXAMPLES::\n\n sage: v = QQ.valuation(2)\n sage: R.<x> = QQ[]\n sage: v = GaussValuation(R, v)\n sage: w = v.augmentation(x, 1337); w # indirect doctest\n [ Gauss valuat... |
class MacLaneApproximantNode():
'\n A node in the tree computed by :meth:`DiscreteValuation.mac_lane_approximants`\n\n Leaves in the computation of the tree of approximants\n :meth:`~DiscreteValuation.mac_lane_approximants`. Each vertex consists of a\n tuple ``(v,ef,p,coeffs,vals)`` where ``v`` is an ... |
class DiscretePseudoValuationSpace(UniqueRepresentation, Homset):
'\n The space of discrete pseudo-valuations on ``domain``.\n\n EXAMPLES::\n\n sage: from sage.rings.valuation.valuation_space import DiscretePseudoValuationSpace\n sage: H = DiscretePseudoValuationSpace(QQ)\n sage: QQ.val... |
class ScaleAction(Action):
'\n Action of integers, rationals and the infinity ring on valuations by\n scaling it.\n\n EXAMPLES::\n\n sage: v = QQ.valuation(5)\n sage: from operator import mul\n sage: v.parent().get_action(ZZ, mul, self_on_left=False)\n Left action by Integer R... |
class DiscreteValuationCodomain(UniqueRepresentation, Parent):
'\n The codomain of discrete valuations, the rational numbers extended by\n `\\pm\\infty`.\n\n EXAMPLES::\n\n sage: from sage.rings.valuation.value_group import DiscreteValuationCodomain\n sage: C = DiscreteValuationCodomain(); ... |
class DiscreteValueGroup(UniqueRepresentation, Parent):
'\n The value group of a discrete valuation, an additive subgroup of `\\QQ`\n generated by ``generator``.\n\n INPUT:\n\n - ``generator`` -- a rational number\n\n .. NOTE::\n\n We do not rely on the functionality provided by additive abe... |
class DiscreteValueSemigroup(UniqueRepresentation, Parent):
'\n The value semigroup of a discrete valuation, an additive subsemigroup of\n `\\QQ` generated by ``generators``.\n\n INPUT:\n\n - ``generators`` -- rational numbers\n\n EXAMPLES::\n\n sage: from sage.rings.valuation.value_group im... |
class SandpileExamples():
'\n Some examples of sandpiles.\n\n Here are the available examples; you can also type\n ``sandpiles.`` and hit tab to get a list:\n\n - :meth:`Complete`\n - :meth:`Cycle`\n - :meth:`Diamond`\n - :meth:`Grid`\n - :meth:`House`\n\n EXAMPLES::\n\n sage: s... |
def _sandpile_help(cls, usage, verbose=True):
'\n Prints help text for classes in this module; see the ``help()`` methods on\n individual classes in this module for example usage.\n '
from sage.misc.sagedoc import detex
methods = []
for attr in sorted(vars(cls)):
if (attr[0] != '_'):
... |
class Sandpile(DiGraph):
"\n Class for Dhar's abelian sandpile model.\n "
@staticmethod
def version():
'\n The version number of Sage Sandpiles.\n\n OUTPUT:\n\n string\n\n\n EXAMPLES::\n\n sage: Sandpile.version()\n Sage Sandpiles Version 2.... |
class SandpileConfig(dict):
'\n Class for configurations on a sandpile.\n '
@staticmethod
def help(verbose=True):
'\n List of SandpileConfig methods. If ``verbose``, include short descriptions.\n\n INPUT:\n\n ``verbose`` -- (default: ``True``) boolean\n\n OUTPUT... |
class SandpileDivisor(dict):
'\n Class for divisors on a sandpile.\n '
@staticmethod
def help(verbose=True):
'\n List of SandpileDivisor methods. If ``verbose``, include short descriptions.\n\n INPUT:\n\n ``verbose`` -- (default: ``True``) boolean\n\n OUTPUT:\n\... |
def sandlib(selector=None):
"\n Returns the sandpile identified by ``selector``. If no argument is\n given, a description of the sandpiles in the sandlib is printed.\n\n INPUT:\n\n ``selector`` -- (optional) identifier or None\n\n OUTPUT:\n\n sandpile or description\n\n EXAMPLES::\n\n ... |
def triangle_sandpile(n):
'\n A triangular sandpile. Each nonsink vertex has out-degree six. The\n vertices on the boundary of the triangle are connected to the sink.\n\n INPUT:\n\n ``n`` -- integer\n\n OUTPUT:\n\n Sandpile\n\n EXAMPLES::\n\n sage: from sage.sandpiles.sandpile import... |
def aztec_sandpile(n):
'\n The aztec diamond graph.\n\n INPUT:\n\n ``n`` -- integer\n\n OUTPUT:\n\n dictionary for the aztec diamond graph\n\n EXAMPLES::\n\n sage: from sage.sandpiles.sandpile import aztec_sandpile\n sage: T = aztec_sandpile(2)\n sage: sorted(len(v) for u, v... |
def glue_graphs(g, h, glue_g, glue_h):
"\n Glue two graphs together.\n\n INPUT:\n\n - ``g``, ``h`` -- dictionaries for directed multigraphs\n\n - ``glue_h``, ``glue_g`` -- dictionaries for a vertex\n\n OUTPUT:\n\n dictionary for a directed multigraph\n\n\n EXAMPLES::\n\n sage: from s... |
def firing_graph(S, eff):
'\n Creates a digraph with divisors as vertices and edges between two divisors\n `D` and `E` if firing a single vertex in `D` gives `E`.\n\n INPUT:\n\n ``S`` -- Sandpile\n\n ``eff`` -- list of divisors\n\n OUTPUT:\n\n DiGraph\n\n EXAMPLES::\n\n sage: S = sa... |
def parallel_firing_graph(S, eff):
'\n Creates a digraph with divisors as vertices and edges between two divisors\n `D` and `E` if firing all unstable vertices in `D` gives `E`.\n\n INPUT:\n\n ``S`` -- Sandpile\n\n ``eff`` -- list of divisors\n\n OUTPUT:\n\n DiGraph\n\n EXAMPLES::\n\n ... |
def admissible_partitions(S, k):
'\n The partitions of the vertices of `S` into `k` parts, each of which is\n connected.\n\n INPUT:\n\n ``S`` -- Sandpile\n\n ``k`` -- integer\n\n OUTPUT:\n\n list of partitions\n\n EXAMPLES::\n\n sage: from sage.sandpiles.sandpile import admissible_p... |
def partition_sandpile(S, p):
'\n Each set of vertices in `p` is regarded as a single vertex, with and edge\n between `A` and `B` if some element of `A` is connected by an edge to some\n element of `B` in `S`.\n\n INPUT:\n\n ``S`` -- Sandpile\n\n ``p`` -- partition of the vertices of ``S``\n\n ... |
def min_cycles(G, v):
"\n Minimal length cycles in the digraph `G` starting at vertex `v`.\n\n INPUT:\n\n - ``G`` -- DiGraph\n\n - ``v`` -- vertex of ``G``\n\n OUTPUT:\n\n list of lists of vertices\n\n EXAMPLES::\n\n sage: from sage.sandpiles.sandpile import min_cycles, sandlib\n ... |
def wilmes_algorithm(M):
'\n Computes an integer matrix `L` with the same integer row span as `M` and\n such that `L` is the reduced Laplacian of a directed multigraph.\n\n INPUT:\n\n ``M`` -- square integer matrix of full rank\n\n OUTPUT:\n\n integer matrix (``L``)\n\n EXAMPLES::\n\n ... |
def solve(F, converter=None, solver=None, n=1, target_variables=None, **kwds):
'\n Solve system of Boolean polynomials ``F`` by solving the\n SAT-problem -- produced by ``converter`` -- using ``solver``.\n\n INPUT:\n\n - ``F`` - a sequence of Boolean polynomials\n\n - ``n`` - number of solutions to... |
def learn(F, converter=None, solver=None, max_learnt_length=3, interreduction=False, **kwds):
"\n Learn new polynomials by running SAT-solver ``solver`` on\n SAT-instance produced by ``converter`` from ``F``.\n\n INPUT:\n\n - ``F`` - a sequence of Boolean polynomials\n\n - ``converter`` - an ANF to... |
class ANF2CNFConverter():
pass
|
class CNFEncoder(ANF2CNFConverter):
"\n ANF to CNF Converter using a Dense/Sparse Strategy. This converter distinguishes two classes of\n polynomials.\n\n 1. Sparse polynomials are those with at most ``max_vars_sparse`` variables. Those are converted\n using reduced truth-tables based on PolyBoRi's in... |
class CryptoMiniSat(SatSolver):
'\n CryptoMiniSat Solver.\n\n INPUT:\n\n - ``verbosity`` -- an integer between 0 and 15 (default: 0). Verbosity.\n\n - ``confl_limit`` -- an integer (default: ``None``). Abort after this many\n conflicts. If set to ``None``, never aborts.\n\n - ``threads`` -- an... |
class DIMACS(SatSolver):
"\n Generic DIMACS Solver.\n\n .. note::\n\n Usually, users won't have to use this class directly but some\n class which inherits from this class.\n\n .. automethod:: __init__\n .. automethod:: __call__\n "
command = ''
def __init__(self, command=None... |
class RSat(DIMACS):
"\n An instance of the RSat solver.\n\n For information on RSat see: http://reasoning.cs.ucla.edu/rsat/\n\n EXAMPLES::\n\n sage: from sage.sat.solvers import RSat\n sage: solver = RSat()\n sage: solver\n DIMACS Solver: 'rsat {input} -v -s'\n\n When the p... |
class Glucose(DIMACS):
"\n An instance of the Glucose solver.\n\n For information on Glucose see: http://www.labri.fr/perso/lsimon/glucose/\n\n EXAMPLES::\n\n sage: from sage.sat.solvers import Glucose\n sage: solver = Glucose()\n sage: solver\n DIMACS Solver: 'glucose -verb=0... |
class GlucoseSyrup(DIMACS):
"\n An instance of the Glucose-syrup parallel solver.\n\n For information on Glucose see: http://www.labri.fr/perso/lsimon/glucose/\n\n EXAMPLES::\n\n sage: from sage.sat.solvers import GlucoseSyrup\n sage: solver = GlucoseSyrup()\n sage: solver\n D... |
class Kissat(DIMACS):
'\n An instance of the Kissat SAT solver\n\n For information on Kissat see: http://fmv.jku.at/kissat/\n\n EXAMPLES::\n\n sage: from sage.sat.solvers import Kissat\n sage: solver = Kissat()\n sage: solver\n DIMACS Solver: \'kissat -q {input}\'\n\n When ... |
class PicoSAT(SatSolver):
'\n PicoSAT Solver.\n\n INPUT:\n\n - ``verbosity`` -- an integer between 0 and 2 (default: 0); verbosity\n\n - ``prop_limit`` -- an integer (default: 0); the propagation limit\n\n EXAMPLES::\n\n sage: from sage.sat.solvers.picosat import PicoSAT\n sage: solve... |
class SatLP(SatSolver):
def __init__(self, solver=None, verbose=0, *, integrality_tolerance=0.001):
'\n Initializes the instance\n\n INPUT:\n\n - ``solver`` -- (default: ``None``) Specify a Mixed Integer Linear Programming\n (MILP) solver to be used. If set to ``None``, the ... |
class SchemeHomset_points_spec(SchemeHomset_generic):
'\n Set of rational points of an affine variety.\n\n INPUT:\n\n See :class:`SchemeHomset_generic`.\n\n EXAMPLES::\n\n sage: from sage.schemes.affine.affine_homset import SchemeHomset_points_spec\n sage: SchemeHomset_points_spec(Spec(Q... |
class SchemeHomset_polynomial_affine_space(SchemeHomset_generic):
'\n Set of morphisms between affine spaces defined by polynomials.\n\n EXAMPLES::\n\n sage: A.<x,y> = AffineSpace(2, QQ)\n sage: Hom(A, A)\n Set of morphisms\n From: Affine Space of dimension 2 over Rational Fiel... |
class SchemeHomset_points_affine(SchemeHomset_points):
'\n Set of rational points of an affine variety.\n\n INPUT:\n\n See :class:`SchemeHomset_generic`.\n\n EXAMPLES::\n\n sage: from sage.schemes.affine.affine_homset import SchemeHomset_points_affine\n sage: SchemeHomset_points_affine(S... |
class SchemeMorphism_polynomial_affine_space(SchemeMorphism_polynomial):
'\n A morphism of schemes determined by rational functions.\n\n EXAMPLES::\n\n sage: RA.<x,y> = QQ[]\n sage: A2 = AffineSpace(RA)\n sage: RP.<u,v,w> = QQ[]\n sage: P2 = ProjectiveSpace(RP)\n sage: H =... |
class SchemeMorphism_polynomial_affine_space_field(SchemeMorphism_polynomial_affine_space):
@cached_method
def weil_restriction(self):
'\n Compute the Weil restriction of this morphism over some extension field.\n\n If the field is a finite field, then this computes\n the Weil re... |
class SchemeMorphism_polynomial_affine_space_finite_field(SchemeMorphism_polynomial_affine_space_field):
def _fast_eval(self, x):
'\n Evaluate affine morphism at point described by ``x``.\n\n EXAMPLES::\n\n sage: P.<x,y,z> = AffineSpace(GF(7), 3)\n sage: H = Hom(P, P)\... |
class SchemeMorphism_polynomial_affine_subscheme_field(SchemeMorphism_polynomial_affine_space_field):
'\n Morphisms from subschemes of affine spaces defined over fields.\n '
@cached_method
def representatives(self):
'\n Return all maps representing the same rational map as by this ma... |
class SchemeMorphism_point_affine(SchemeMorphism_point):
'\n A rational point on an affine scheme.\n\n INPUT:\n\n - ``X`` -- a subscheme of an ambient affine space over a ring `R`\n\n - ``v`` -- a list/tuple/iterable of coordinates in `R`\n\n - ``check`` -- boolean (optional, default:``True``); whe... |
class SchemeMorphism_point_affine_field(SchemeMorphism_point_affine):
def __hash__(self):
'\n Compute the hash value of this affine point.\n\n EXAMPLES::\n\n sage: A.<x,y> = AffineSpace(QQ, 2)\n sage: X = A.subscheme(x - y)\n sage: hash(X([1, 1])) == hash((1... |
class SchemeMorphism_point_affine_finite_field(SchemeMorphism_point_affine_field):
def __hash__(self):
"\n Returns the integer hash of the point.\n\n OUTPUT: Integer.\n\n EXAMPLES::\n\n sage: P.<x,y,z> = AffineSpace(GF(5), 3)\n sage: hash(P(2, 1, 2))\n ... |
def enum_affine_rational_field(X, B):
'\n Enumerates affine rational points on scheme ``X`` up to bound ``B``.\n\n INPUT:\n\n - ``X`` - a scheme or set of abstract rational points of a scheme.\n - ``B`` - a positive integer bound.\n\n OUTPUT:\n\n - a list containing the affine points of ``X`` ... |
def enum_affine_number_field(X, **kwds):
"\n Enumerates affine points on scheme ``X`` defined over a number field. Simply checks all of the\n points of absolute height up to ``B`` and adds those that are on the scheme to the list.\n\n This algorithm computes 2 lists: L containing elements x in `K` such t... |
def enum_affine_finite_field(X):
'\n Enumerates affine points on scheme ``X`` defined over a finite field.\n\n INPUT:\n\n - ``X`` - a scheme defined over a finite field or a set of abstract\n rational points of such a scheme.\n\n OUTPUT:\n\n - a list containing the affine points of ``X`` over... |
def is_AffineSpace(x) -> bool:
"\n Return ``True`` if ``x`` is an affine space.\n\n EXAMPLES::\n\n sage: from sage.schemes.affine.affine_space import is_AffineSpace\n sage: is_AffineSpace(AffineSpace(5, names='x'))\n True\n sage: is_AffineSpace(AffineSpace(5, GF(9, 'alpha'), name... |
def AffineSpace(n, R=None, names=None, ambient_projective_space=None, default_embedding_index=None):
"\n Return affine space of dimension ``n`` over the ring ``R``.\n\n EXAMPLES:\n\n The dimension and ring can be given in either order::\n\n sage: AffineSpace(3, QQ, 'x')\n Affine Space of di... |
class AffineSpace_generic(AmbientSpace, AffineScheme):
"\n Affine space of dimension `n` over the ring `R`.\n\n EXAMPLES::\n\n sage: X.<x,y,z> = AffineSpace(3, QQ)\n sage: X.base_scheme()\n Spectrum of Rational Field\n sage: X.base_ring()\n Rational Field\n sage: X.... |
class AffineSpace_field(AffineSpace_generic):
def _point(self, *args, **kwds):
'\n Construct a point.\n\n For internal use only. See :mod:`morphism` for details.\n\n TESTS::\n\n sage: P2.<x,y,z> = AffineSpace(3, GF(3))\n sage: point_homset = P2._point_homset(Spe... |
class AffineSpace_finite_field(AffineSpace_field):
def _point(self, *args, **kwds):
'\n Construct a point.\n\n For internal use only. See :mod:`morphism` for details.\n\n TESTS::\n\n sage: P2.<x,y,z> = AffineSpace(3, GF(3))\n sage: point_homset = P2._point_homse... |
class AlgebraicScheme_subscheme_affine(AlgebraicScheme_subscheme):
'\n An algebraic subscheme of affine space.\n\n INPUT:\n\n - ``A`` -- ambient affine space\n\n - ``polynomials`` -- single polynomial, ideal or iterable of defining\n polynomials\n\n EXAMPLES::\n\n sage: A3.<x, y, z> = A... |
class AlgebraicScheme_subscheme_affine_field(AlgebraicScheme_subscheme_affine):
'\n Algebraic subschemes of projective spaces defined over fields.\n '
def _morphism(self, *args, **kwds):
'\n Construct a morphism determined by action on points of ``self``.\n\n TESTS::\n\n ... |
class Berkovich_Element(Element):
'\n The parent class for any element of a Berkovich space.\n '
pass
|
class Berkovich_Element_Cp(Berkovich_Element):
'\n The abstract parent class for any element of Berkovich space over `\\CC_p`.\n\n This class should never be instantiated, instead use :class:`Berkovich_Element_Cp_Affine`\n or :class:`Berkovich_Element_Cp_Projective`.\n\n EXAMPLES::\n\n sage: B ... |
class Berkovich_Element_Cp_Affine(Berkovich_Element_Cp):
'\n Element class of the Berkovich affine line over `\\CC_p`.\n\n Elements are categorized into four types, represented by specific data:\n\n - Type I points are represented by a center in the ``base`` of the parent Berkovich space,\n which is... |
class Berkovich_Element_Cp_Projective(Berkovich_Element_Cp):
'\n Element class of the Berkovich projective line over `\\CC_p`.\n\n Elements are categorized into four types, represented by specific data:\n\n - Type I points are represented by a center in the ``base`` of the parent Berkovich space,\n ... |
def is_Berkovich(space) -> bool:
'\n Check if ``space`` is a Berkovich space.\n\n OUTPUT:\n\n - ``True`` if ``space`` is a Berkovich space.\n - ``False`` otherwise.\n\n EXAMPLES::\n\n sage: B = Berkovich_Cp_Projective(3)\n sage: from sage.schemes.berkovich.berkovich_space import is_Be... |
def is_Berkovich_Cp(space) -> bool:
'\n Check if ``space`` is a Berkovich space over ``Cp``.\n\n OUTPUT:\n\n - ``True`` if ``space`` is a Berkovich space over ``Cp``.\n - ``False`` otherwise.\n\n EXAMPLES::\n\n sage: B = Berkovich_Cp_Projective(3)\n sage: from sage.schemes.berkovich.b... |
class Berkovich(UniqueRepresentation, Parent):
'\n The parent class for any Berkovich space\n '
pass
|
class Berkovich_Cp(Berkovich):
'\n Abstract parent class for Berkovich space over ``Cp``.\n '
def residue_characteristic(self):
'\n The residue characteristic of the ``base``.\n\n EXAMPLES::\n\n sage: B = Berkovich_Cp_Projective(3)\n sage: B.prime()\n ... |
class Berkovich_Cp_Affine(Berkovich_Cp):
'\n The Berkovich affine line over `\\CC_p`.\n\n The Berkovich affine line is the set of seminorms on `\\CC_p[x]`,\n with the weakest topology such that the map `| \\cdot | \\to |f|` is continuous\n for all `f \\in \\CC_p[x]`.\n\n We can represent the Berkov... |
class Berkovich_Cp_Projective(Berkovich_Cp):
'\n The Berkovich projective line over `\\CC_p`.\n\n The Berkovich projective line is the one-point compactification\n of the Berkovich affine line.\n\n We can represent the Berkovich projective line in two separate ways:\n either using a p-adic field to... |
class AffineCurve(Curve_generic, AlgebraicScheme_subscheme_affine):
'\n Affine curves.\n\n EXAMPLES::\n\n sage: R.<v> = QQ[]\n sage: K.<u> = NumberField(v^2 + 3) # needs sage.rings.number_field\n sage: A.<x,y,z> = AffineSpace(K, 3) ... |
class AffinePlaneCurve(AffineCurve):
'\n Affine plane curves.\n '
def __init__(self, A, f):
'\n Initialize.\n\n EXAMPLES::\n\n sage: A.<x,y> = AffineSpace(QQ, 2)\n sage: C = Curve([x^3 - y^2], A); C\n Affine Plane Curve over Rational Field defined ... |
class AffineCurve_field(AffineCurve, AlgebraicScheme_subscheme_affine_field):
'\n Affine curves over fields.\n '
_point = AffineCurvePoint_field
def __init__(self, A, X):
'\n Initialize.\n\n EXAMPLES::\n\n sage: R.<v> = QQ[]\n sage: K.<u> = NumberField(v^... |
class AffinePlaneCurve_field(AffinePlaneCurve, AffineCurve_field):
'\n Affine plane curves over fields.\n '
_point = AffinePlaneCurvePoint_field
@cached_method
def fundamental_group(self, simplified=True, puiseux=False):
"\n Return a presentation of the fundamental group of the c... |
class AffinePlaneCurve_finite_field(AffinePlaneCurve_field):
'\n Affine plane curves over finite fields.\n '
_point = AffinePlaneCurvePoint_finite_field
def riemann_roch_basis(self, D):
'\n Return a basis of the Riemann-Roch space of the divisor ``D``.\n\n This interfaces with... |
class IntegralAffineCurve(AffineCurve_field):
'\n Base class for integral affine curves.\n '
_point = IntegralAffineCurvePoint
_closed_point = IntegralAffineCurveClosedPoint
def function_field(self):
'\n Return the function field of the curve.\n\n EXAMPLES::\n\n ... |
class IntegralAffineCurve_finite_field(IntegralAffineCurve):
'\n Integral affine curves.\n\n INPUT:\n\n - ``A`` -- an ambient space in which the curve lives\n\n - ``X`` -- list of polynomials that define the curve\n\n EXAMPLES::\n\n sage: A.<x,y,z> = AffineSpace(GF(11), 3)\n sage: C =... |
class IntegralAffinePlaneCurve(IntegralAffineCurve, AffinePlaneCurve_field):
_point = IntegralAffinePlaneCurvePoint
|
class IntegralAffinePlaneCurve_finite_field(AffinePlaneCurve_finite_field, IntegralAffineCurve_finite_field):
'\n Integral affine plane curve over a finite field.\n\n EXAMPLES::\n\n sage: # needs sage.rings.finite_rings\n sage: A.<x,y> = AffineSpace(GF(8), 2)\n sage: C = Curve(x^5 + y^5... |
class CurveClosedPoint(SchemeTopologicalPoint_prime_ideal):
'\n Base class of closed points of curves.\n '
pass
|
class IntegralCurveClosedPoint(CurveClosedPoint):
'\n Closed points of integral curves.\n\n INPUT:\n\n - ``curve`` -- the curve to which the closed point belongs\n\n - ``prime_ideal`` -- a prime ideal\n\n - ``degree`` -- degree of the closed point\n\n EXAMPLES::\n\n sage: # needs sage.ri... |
class IntegralAffineCurveClosedPoint(IntegralCurveClosedPoint):
'\n Closed points of affine curves.\n '
def rational_point(self):
'\n Return the rational point if this closed point is of degree `1`.\n\n EXAMPLES::\n\n sage: # needs sage.rings.finite_rings\n s... |
class IntegralProjectiveCurveClosedPoint(IntegralCurveClosedPoint):
'\n Closed points of projective plane curves.\n '
def rational_point(self):
'\n Return the rational point if this closed point is of degree `1`.\n\n EXAMPLES::\n\n sage: # needs sage.rings.finite_rings\... |
def _is_irreducible_and_reduced(F) -> bool:
'\n Check if the polynomial F is irreducible and reduced.\n\n TESTS::\n\n sage: R.<x,y> = QQ[]\n sage: F = x^2 + y^2\n sage: from sage.schemes.curves.constructor import _is_irreducible_and_reduced\n sage: _is_irreducible_and_reduced(F)\... |
def Curve(F, A=None):
"\n Return the plane or space curve defined by ``F``, where ``F`` can be either\n a multivariate polynomial, a list or tuple of polynomials, or an algebraic\n scheme.\n\n If no ambient space is passed in for ``A``, and if ``F`` is not an\n algebraic scheme, a new ambient space... |
class Curve_generic(AlgebraicScheme_subscheme):
'\n Generic curve class.\n\n EXAMPLES::\n\n sage: A.<x,y,z> = AffineSpace(QQ, 3)\n sage: C = Curve([x - y, z - 2])\n sage: loads(C.dumps()) == C\n True\n '
def _repr_(self):
'\n Return a string representation ... |
class ProjectiveCurvePoint_field(SchemeMorphism_point_projective_field):
'\n Point of a projective curve over a field.\n '
def is_singular(self):
'\n Return whether this point is a singular point of the projective curve it is on.\n\n EXAMPLES::\n\n sage: P.<x,y,z,w> = P... |
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