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""" Affine curves.
EXAMPLES:
We can construct curves in either an affine plane::
sage: A.<x,y> = AffineSpace(QQ, 2) sage: C = Curve([y - x^2], A); C Affine Plane Curve over Rational Field defined by -x^2 + y
or in higher dimensional affine space::
sage: A.<x,y,z,w> = AffineSpace(QQ, 4) sage: C = Curve([y - x^2, z - w^3, w - y^4], A); C Affine Curve over Rational Field defined by -x^2 + y, -w^3 + z, -y^4 + w
AUTHORS:
- William Stein (2005-11-13)
- David Joyner (2005-11-13)
- David Kohel (2006-01)
- Grayson Jorgenson (2016-8) """ #***************************************************************************** # Copyright (C) 2005 William Stein <wstein@gmail.com> # # Distributed under the terms of the GNU General Public License (GPL) # # The full text of the GPL is available at: # # http://www.gnu.org/licenses/ #***************************************************************************** from __future__ import absolute_import
from sage.arith.misc import binomial from sage.categories.fields import Fields from sage.categories.finite_fields import FiniteFields from copy import copy from sage.categories.homset import Hom, End from sage.categories.number_fields import NumberFields from sage.interfaces.all import singular import sage.libs.singular
from sage.misc.all import add
from sage.rings.all import degree_lowest_rational_function
from sage.rings.number_field.number_field import NumberField from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing from sage.rings.qqbar import number_field_elements_from_algebraics, QQbar from sage.rings.rational_field import is_RationalField from sage.schemes.affine.affine_space import (AffineSpace, is_AffineSpace) from . import point
from sage.schemes.affine.affine_subscheme import AlgebraicScheme_subscheme_affine
from sage.schemes.affine.affine_space import AffineSpace, is_AffineSpace from sage.schemes.projective.projective_space import ProjectiveSpace
from .curve import Curve_generic
class AffineCurve(Curve_generic, AlgebraicScheme_subscheme_affine):
_point = point.AffineCurvePoint_field
def _repr_type(self): r""" Return a string representation of the type of this curve.
EXAMPLES::
sage: A.<x,y,z,w> = AffineSpace(QQ, 4) sage: C = Curve([x - y, z - w, w - x], A) sage: C._repr_type() 'Affine' """
def __init__(self, A, X): r""" Initialization function.
EXAMPLES::
sage: R.<v> = QQ[] sage: K.<u> = NumberField(v^2 + 3) sage: A.<x,y,z> = AffineSpace(K, 3) sage: C = Curve([z - u*x^2, y^2], A); C Affine Curve over Number Field in u with defining polynomial v^2 + 3 defined by (-u)*x^2 + z, y^2
::
sage: A.<x,y,z> = AffineSpace(GF(7), 3) sage: C = Curve([x^2 - z, z - 8*x], A); C Affine Curve over Finite Field of size 7 defined by x^2 - z, -x + z """ raise TypeError("A (=%s) must be an affine space"%A)
def projective_closure(self, i=0, PP=None): r""" Return the projective closure of this affine curve.
INPUT:
- ``i`` -- (default: 0) the index of the affine coordinate chart of the projective space that the affine ambient space of this curve embeds into.
- ``PP`` -- (default: None) ambient projective space to compute the projective closure in. This is constructed if it is not given.
OUTPUT:
- a curve in projective space.
EXAMPLES::
sage: A.<x,y,z> = AffineSpace(QQ, 3) sage: C = Curve([y-x^2,z-x^3], A) sage: C.projective_closure() Projective Curve over Rational Field defined by x1^2 - x0*x2, x1*x2 - x0*x3, x2^2 - x1*x3
::
sage: A.<x,y,z> = AffineSpace(QQ, 3) sage: C = Curve([y - x^2, z - x^3], A) sage: C.projective_closure() Projective Curve over Rational Field defined by x1^2 - x0*x2, x1*x2 - x0*x3, x2^2 - x1*x3
::
sage: A.<x,y> = AffineSpace(CC, 2) sage: C = Curve(y - x^3 + x - 1, A) sage: C.projective_closure(1) Projective Plane Curve over Complex Field with 53 bits of precision defined by x0^3 - x0*x1^2 + x1^3 - x1^2*x2
::
sage: A.<x,y> = AffineSpace(QQ, 2) sage: P.<u,v,w> = ProjectiveSpace(QQ, 2) sage: C = Curve([y - x^2], A) sage: C.projective_closure(1, P).ambient_space() == P True """
def projection(self, indices, AS=None): r""" Return the projection of this curve onto the coordinates specified by ``indices``.
INPUT:
- ``indices`` -- a list or tuple of distinct integers specifying the indices of the coordinates to use in the projection. Can also be a list or tuple consisting of variables of the coordinate ring of the ambient space of this curve. If integers are used to specify the coordinates, 0 denotes the first coordinate. The length of ``indices`` must be between two and one less than the dimension of the ambient space of this curve, inclusive.
- ``AS`` -- (default: None) the affine space the projected curve will be defined in. This space must be defined over the same base field as this curve, and must have dimension equal to the length of ``indices``. This space is constructed if not specified.
OUTPUT:
- a tuple consisting of two elements: a scheme morphism from this curve to affine space of dimension equal to the number of coordinates specified in ``indices``, and the affine subscheme that is the image of that morphism. If the image is a curve, the second element of the tuple will be a curve.
EXAMPLES::
sage: A.<x,y,z> = AffineSpace(QQ, 3) sage: C = Curve([y^7 - x^2 + x^3 - 2*z, z^2 - x^7 - y^2], A) sage: C.projection([0,1]) (Scheme morphism: From: Affine Curve over Rational Field defined by y^7 + x^3 - x^2 - 2*z, -x^7 - y^2 + z^2 To: Affine Space of dimension 2 over Rational Field Defn: Defined on coordinates by sending (x, y, z) to (x, y), Affine Plane Curve over Rational Field defined by x1^14 + 2*x0^3*x1^7 - 2*x0^2*x1^7 - 4*x0^7 + x0^6 - 2*x0^5 + x0^4 - 4*x1^2) sage: C.projection([0,1,3,4]) Traceback (most recent call last): ... ValueError: (=[0, 1, 3, 4]) must be a list or tuple of length between 2 and (=2), inclusive
::
sage: A.<x,y,z,w> = AffineSpace(QQ, 4) sage: C = Curve([x - 2, y - 3, z - 1], A) sage: B.<a,b,c> = AffineSpace(QQ, 3) sage: C.projection([0,1,2], AS=B) (Scheme morphism: From: Affine Curve over Rational Field defined by x - 2, y - 3, z - 1 To: Affine Space of dimension 3 over Rational Field Defn: Defined on coordinates by sending (x, y, z, w) to (x, y, z), Closed subscheme of Affine Space of dimension 3 over Rational Field defined by: c - 1, b - 3, a - 2)
::
sage: A.<x,y,z,w,u> = AffineSpace(GF(11), 5) sage: C = Curve([x^3 - 5*y*z + u^2, x - y^2 + 3*z^2, w^2 + 2*u^3*y, y - u^2 + z*x], A) sage: B.<a,b,c> = AffineSpace(GF(11), 3) sage: proj1 = C.projection([1,2,4], AS=B) sage: proj1 (Scheme morphism: From: Affine Curve over Finite Field of size 11 defined by x^3 - 5*y*z + u^2, -y^2 + 3*z^2 + x, 2*y*u^3 + w^2, x*z - u^2 + y To: Affine Space of dimension 3 over Finite Field of size 11 Defn: Defined on coordinates by sending (x, y, z, w, u) to (y, z, u), Affine Curve over Finite Field of size 11 defined by a^2*b - 3*b^3 - c^2 + a, c^6 - 5*a*b^4 + b^3*c^2 - 3*a*c^4 + 3*a^2*c^2 - a^3, a^2*c^4 - 3*b^2*c^4 - 2*a^3*c^2 - 5*a*b^2*c^2 + a^4 - 5*a*b^3 + 2*b^4 + b^2*c^2 - 3*b*c^2 + 3*a*b, a^4*c^2 + 2*b^4*c^2 - a^5 - 2*a*b^4 + 5*b*c^4 + a*b*c^2 - 5*a*b^2 + 4*b^3 + b*c^2 + 5*c^2 - 5*a, a^6 - 5*b^6 - 5*b^3*c^2 + 5*a*b^3 + 2*c^4 - 4*a*c^2 + 2*a^2 - 5*a*b + c^2) sage: proj1[1].ambient_space() is B True sage: proj2 = C.projection([1,2,4]) sage: proj2[1].ambient_space() is B False sage: C.projection([1,2,3,5], AS=B) Traceback (most recent call last): ... TypeError: (=Affine Space of dimension 3 over Finite Field of size 11) must have dimension (=4)
::
sage: A.<x,y,z,w> = AffineSpace(QQ, 4) sage: C = A.curve([x*y - z^3, x*z - w^3, w^2 - x^3]) sage: C.projection([y,z]) (Scheme morphism: From: Affine Curve over Rational Field defined by -z^3 + x*y, -w^3 + x*z, -x^3 + w^2 To: Affine Space of dimension 2 over Rational Field Defn: Defined on coordinates by sending (x, y, z, w) to (y, z), Affine Plane Curve over Rational Field defined by x1^23 - x0^7*x1^4) sage: B.<x,y,z> = AffineSpace(QQ, 3) sage: C.projection([x,y,z], AS=B) (Scheme morphism: From: Affine Curve over Rational Field defined by -z^3 + x*y, -w^3 + x*z, -x^3 + w^2 To: Affine Space of dimension 3 over Rational Field Defn: Defined on coordinates by sending (x, y, z, w) to (x, y, z), Affine Curve over Rational Field defined by z^3 - x*y, x^8 - x*z^2, x^7*z^2 - x*y*z) sage: C.projection([y,z,z]) Traceback (most recent call last): ... ValueError: (=[y, z, z]) must be a list or tuple of distinct indices or variables """ raise TypeError("this curve is already a plane curve") raise TypeError("this curve must be defined over a field") raise TypeError("(=%s) must be an affine space"%AS) raise TypeError("(=%s) must be defined over the same base field as this curve"%AS) else: raise ValueError("index values must be between 0 and one minus the dimension of the ambient space " \ "of this curve") # construct the projection map else: # compute the image via elimination
def plane_projection(self, AP=None): r""" Return a projection of this curve into an affine plane so that the image of the projection is a plane curve.
INPUT:
- ``AP`` -- (default: None) the affine plane to project this curve into. This space must be defined over the same base field as this curve, and must have dimension two. This space will be constructed if not specified.
OUTPUT:
- a tuple consisting of two elements: a scheme morphism from this curve into an affine plane, and the plane curve that defines the image of that morphism.
EXAMPLES::
sage: A.<x,y,z,w> = AffineSpace(QQ, 4) sage: C = Curve([x^2 - y*z*w, z^3 - w, w + x*y - 3*z^3], A) sage: C.plane_projection() (Scheme morphism: From: Affine Curve over Rational Field defined by -y*z*w + x^2, z^3 - w, -3*z^3 + x*y + w To: Affine Space of dimension 2 over Rational Field Defn: Defined on coordinates by sending (x, y, z, w) to (x, y), Affine Plane Curve over Rational Field defined by x0^2*x1^7 - 16*x0^4)
::
sage: R.<a> = QQ[] sage: K.<b> = NumberField(a^2 + 2) sage: A.<x,y,z> = AffineSpace(K, 3) sage: C = A.curve([x - b, y - 2]) sage: B.<a,b> = AffineSpace(K, 2) sage: proj1 = C.plane_projection(AP=B) sage: proj1 (Scheme morphism: From: Affine Curve over Number Field in b with defining polynomial a^2 + 2 defined by x + (-b), y - 2 To: Affine Space of dimension 2 over Number Field in b with defining polynomial a^2 + 2 Defn: Defined on coordinates by sending (x, y, z) to (x, z), Affine Plane Curve over Number Field in b with defining polynomial a^2 + 2 defined by a + (-b)) sage: proj1[1].ambient_space() is B True sage: proj2 = C.plane_projection() sage: proj2[1].ambient_space() is B False """ # finds a projection that will have a plane curve as its image # the following iterates over all pairs (i,j) with j > i to test all # possible projections
def blowup(self, P=None): r""" Return the blow up of this affine curve at the point ``P``.
The blow up is described by affine charts. This curve must be irreducible.
INPUT:
- ``P`` -- (default: None) a point on this curve at which to blow up. If ``None``, then ``P`` is taken to be the origin.
OUTPUT:
- a tuple consisting of three elements. The first is a tuple of curves in affine space of the same dimension as the ambient space of this curve, which define the blow up in each affine chart. The second is a tuple of tuples such that the jth element of the ith tuple is the transition map from the ith affine patch to the jth affine patch. Lastly, the third element is a tuple consisting of the restrictions of the projection map from the blow up back to the original curve, restricted to each affine patch. There the ith element will be the projection from the ith affine patch.
EXAMPLES::
sage: A.<x,y> = AffineSpace(QQ, 2) sage: C = Curve([y^2 - x^3], A) sage: C.blowup() ((Affine Plane Curve over Rational Field defined by s1^2 - x, Affine Plane Curve over Rational Field defined by y*s0^3 - 1), ([Scheme endomorphism of Affine Plane Curve over Rational Field defined by s1^2 - x Defn: Defined on coordinates by sending (x, s1) to (x, s1), Scheme morphism: From: Affine Plane Curve over Rational Field defined by s1^2 - x To: Affine Plane Curve over Rational Field defined by y*s0^3 - 1 Defn: Defined on coordinates by sending (x, s1) to (x*s1, 1/s1)], [Scheme morphism: From: Affine Plane Curve over Rational Field defined by y*s0^3 - 1 To: Affine Plane Curve over Rational Field defined by s1^2 - x Defn: Defined on coordinates by sending (y, s0) to (y*s0, 1/s0), Scheme endomorphism of Affine Plane Curve over Rational Field defined by y*s0^3 - 1 Defn: Defined on coordinates by sending (y, s0) to (y, s0)]), (Scheme morphism: From: Affine Plane Curve over Rational Field defined by s1^2 - x To: Affine Plane Curve over Rational Field defined by -x^3 + y^2 Defn: Defined on coordinates by sending (x, s1) to (x, x*s1), Scheme morphism: From: Affine Plane Curve over Rational Field defined by y*s0^3 - 1 To: Affine Plane Curve over Rational Field defined by -x^3 + y^2 Defn: Defined on coordinates by sending (y, s0) to (y*s0, y)))
::
sage: K.<a> = QuadraticField(2) sage: A.<x,y,z> = AffineSpace(K, 3) sage: C = Curve([y^2 - a*x^5, x - z], A) sage: B = C.blowup() sage: B[0] (Affine Curve over Number Field in a with defining polynomial x^2 - 2 defined by s2 - 1, 2*x^3 + (-a)*s1^2, Affine Curve over Number Field in a with defining polynomial x^2 - 2 defined by s0 - s2, 2*y^3*s2^5 + (-a), Affine Curve over Number Field in a with defining polynomial x^2 - 2 defined by s0 - 1, 2*z^3 + (-a)*s1^2) sage: B[1][0][2] Scheme morphism: From: Affine Curve over Number Field in a with defining polynomial x^2 - 2 defined by s2 - 1, 2*x^3 + (-a)*s1^2 To: Affine Curve over Number Field in a with defining polynomial x^2 - 2 defined by s0 - 1, 2*z^3 + (-a)*s1^2 Defn: Defined on coordinates by sending (x, s1, s2) to (x*s2, 1/s2, s1/s2) sage: B[1][2][0] Scheme morphism: From: Affine Curve over Number Field in a with defining polynomial x^2 - 2 defined by s0 - 1, 2*z^3 + (-a)*s1^2 To: Affine Curve over Number Field in a with defining polynomial x^2 - 2 defined by s2 - 1, 2*x^3 + (-a)*s1^2 Defn: Defined on coordinates by sending (z, s0, s1) to (z*s0, s1/s0, 1/s0) sage: B[2] (Scheme morphism: From: Affine Curve over Number Field in a with defining polynomial x^2 - 2 defined by s2 - 1, 2*x^3 + (-a)*s1^2 To: Affine Curve over Number Field in a with defining polynomial x^2 - 2 defined by (-a)*x^5 + y^2, x - z Defn: Defined on coordinates by sending (x, s1, s2) to (x, x*s1, x*s2), Scheme morphism: From: Affine Curve over Number Field in a with defining polynomial x^2 - 2 defined by s0 - s2, 2*y^3*s2^5 + (-a) To: Affine Curve over Number Field in a with defining polynomial x^2 - 2 defined by (-a)*x^5 + y^2, x - z Defn: Defined on coordinates by sending (y, s0, s2) to (y*s0, y, y*s2), Scheme morphism: From: Affine Curve over Number Field in a with defining polynomial x^2 - 2 defined by s0 - 1, 2*z^3 + (-a)*s1^2 To: Affine Curve over Number Field in a with defining polynomial x^2 - 2 defined by (-a)*x^5 + y^2, x - z Defn: Defined on coordinates by sending (z, s0, s1) to (z*s0, z*s1, z))
::
sage: A.<x,y> = AffineSpace(QQ, 2) sage: C = A.curve((y - 3/2)^3 - (x + 2)^5 - (x + 2)^6) sage: Q = A([-2,3/2]) sage: C.blowup(Q) ((Affine Plane Curve over Rational Field defined by x^3 - s1^3 + 7*x^2 + 16*x + 12, Affine Plane Curve over Rational Field defined by 8*y^3*s0^6 - 36*y^2*s0^6 + 8*y^2*s0^5 + 54*y*s0^6 - 24*y*s0^5 - 27*s0^6 + 18*s0^5 - 8), ([Scheme endomorphism of Affine Plane Curve over Rational Field defined by x^3 - s1^3 + 7*x^2 + 16*x + 12 Defn: Defined on coordinates by sending (x, s1) to (x, s1), Scheme morphism: From: Affine Plane Curve over Rational Field defined by x^3 - s1^3 + 7*x^2 + 16*x + 12 To: Affine Plane Curve over Rational Field defined by 8*y^3*s0^6 - 36*y^2*s0^6 + 8*y^2*s0^5 + 54*y*s0^6 - 24*y*s0^5 - 27*s0^6 + 18*s0^5 - 8 Defn: Defined on coordinates by sending (x, s1) to (x*s1 + 2*s1 + 3/2, 1/s1)], [Scheme morphism: From: Affine Plane Curve over Rational Field defined by 8*y^3*s0^6 - 36*y^2*s0^6 + 8*y^2*s0^5 + 54*y*s0^6 - 24*y*s0^5 - 27*s0^6 + 18*s0^5 - 8 To: Affine Plane Curve over Rational Field defined by x^3 - s1^3 + 7*x^2 + 16*x + 12 Defn: Defined on coordinates by sending (y, s0) to (y*s0 - 3/2*s0 - 2, 1/s0), Scheme endomorphism of Affine Plane Curve over Rational Field defined by 8*y^3*s0^6 - 36*y^2*s0^6 + 8*y^2*s0^5 + 54*y*s0^6 - 24*y*s0^5 - 27*s0^6 + 18*s0^5 - 8 Defn: Defined on coordinates by sending (y, s0) to (y, s0)]), (Scheme morphism: From: Affine Plane Curve over Rational Field defined by x^3 - s1^3 + 7*x^2 + 16*x + 12 To: Affine Plane Curve over Rational Field defined by -x^6 - 13*x^5 - 70*x^4 - 200*x^3 + y^3 - 320*x^2 - 9/2*y^2 - 272*x + 27/4*y - 795/8 Defn: Defined on coordinates by sending (x, s1) to (x, x*s1 + 2*s1 + 3/2), Scheme morphism: From: Affine Plane Curve over Rational Field defined by 8*y^3*s0^6 - 36*y^2*s0^6 + 8*y^2*s0^5 + 54*y*s0^6 - 24*y*s0^5 - 27*s0^6 + 18*s0^5 - 8 To: Affine Plane Curve over Rational Field defined by -x^6 - 13*x^5 - 70*x^4 - 200*x^3 + y^3 - 320*x^2 - 9/2*y^2 - 272*x + 27/4*y - 795/8 Defn: Defined on coordinates by sending (y, s0) to (y*s0 - 3/2*s0 - 2, y)))
::
sage: A.<x,y,z,w> = AffineSpace(QQ, 4) sage: C = A.curve([((x + 1)^2 + y^2)^3 - 4*(x + 1)^2*y^2, y - z, w - 4]) sage: Q = C([-1,0,0,4]) sage: B = C.blowup(Q) sage: B[0] (Affine Curve over Rational Field defined by s3, s1 - s2, x^2*s2^6 + 2*x*s2^6 + 3*x^2*s2^4 + s2^6 + 6*x*s2^4 + 3*x^2*s2^2 + 3*s2^4 + 6*x*s2^2 + x^2 - s2^2 + 2*x + 1, Affine Curve over Rational Field defined by s3, s2 - 1, y^2*s0^6 + 3*y^2*s0^4 + 3*y^2*s0^2 + y^2 - 4*s0^2, Affine Curve over Rational Field defined by s3, s1 - 1, z^2*s0^6 + 3*z^2*s0^4 + 3*z^2*s0^2 + z^2 - 4*s0^2, Closed subscheme of Affine Space of dimension 4 over Rational Field defined by: 1) sage: Q = A([6,2,3,1]) sage: B = C.blowup(Q) Traceback (most recent call last): ... TypeError: (=(6, 2, 3, 1)) must be a point on this curve
::
sage: A.<x,y> = AffineSpace(QuadraticField(-1), 2) sage: C = A.curve([y^2 + x^2]) sage: C.blowup() Traceback (most recent call last): ... TypeError: this curve must be irreducible """ raise TypeError("the base ring of this curve must be a field") # attempt to make the variable names more organized # the convention used here is to have the homogeneous coordinates for the projective component of the # product space the blow up resides in be generated from the letter 's'. The following loop is in place # to prevent conflicts in the names from occurring rf = len(str(A.gens()[i])) # move the defining polynomials of this curve into R # the blow up ideal of A at P is the ideal generated by # (z_i - p_i)*s_j - (z_j - p_j)*s_i for i != j from 0,...,n-1 # in the mixed product space of A^n and P^{n-1} where the z_i are the gens # of A^n, the s_i are the gens for P^{n-1}, and P = (p_1,...,p_n). We describe the # blow up of this curve at P in each affine chart # in this chart, s_i is assumed to be 1 # substitute in z_j = (z_i - p_i)*s_j + p_j for each j != i else: # choose the component of the subscheme defined by these polynomials # that corresponds to the proper transform # patch of blowup in this chart is empty # create the transition maps between the charts # create the restrictions of the projection map
def resolution_of_singularities(self, extend=False): r""" Return a nonsingular model for this affine curve created by blowing up its singular points.
The nonsingular model is given as a collection of affine patches that cover it. If ``extend`` is ``False`` and if the base field is a number field, or if the base field is a finite field, the model returned may have singularities with coordinates not contained in the base field. An error is returned if this curve is already nonsingular, or if it has no singular points over its base field. This curve must be irreducible, and must be defined over a number field or finite field.
INPUT:
- ``extend`` -- (default: False) specifies whether to extend the base field when necessary to find all singular points when this curve is defined over a number field. If ``extend`` is ``False``, then only singularities with coordinates in the base field of this curve will be resolved. However, setting ``extend`` to ``True`` will slow down computations.
OUTPUT:
- a tuple consisting of three elements. The first is a tuple of curves in affine space of the same dimension as the ambient space of this curve, which represent affine patches of the resolution of singularities. The second is a tuple of tuples such that the jth element of the ith tuple is the transition map from the ith patch to the jth patch. Lastly, the third element is a tuple consisting of birational maps from the patches back to the original curve that were created by composing the projection maps generated from the blow up computations. There the ith element will be a map from the ith patch.
EXAMPLES::
sage: A.<x,y> = AffineSpace(QQ, 2) sage: C = Curve([y^2 - x^3], A) sage: C.resolution_of_singularities() ((Affine Plane Curve over Rational Field defined by s1^2 - x, Affine Plane Curve over Rational Field defined by y*s0^3 - 1), ((Scheme endomorphism of Affine Plane Curve over Rational Field defined by s1^2 - x Defn: Defined on coordinates by sending (x, s1) to (x, s1), Scheme morphism: From: Affine Plane Curve over Rational Field defined by s1^2 - x To: Affine Plane Curve over Rational Field defined by y*s0^3 - 1 Defn: Defined on coordinates by sending (x, s1) to (x*s1, 1/s1)), (Scheme morphism: From: Affine Plane Curve over Rational Field defined by y*s0^3 - 1 To: Affine Plane Curve over Rational Field defined by s1^2 - x Defn: Defined on coordinates by sending (y, s0) to (y*s0, 1/s0), Scheme endomorphism of Affine Plane Curve over Rational Field defined by y*s0^3 - 1 Defn: Defined on coordinates by sending (y, s0) to (y, s0))), (Scheme morphism: From: Affine Plane Curve over Rational Field defined by s1^2 - x To: Affine Plane Curve over Rational Field defined by -x^3 + y^2 Defn: Defined on coordinates by sending (x, s1) to (x, x*s1), Scheme morphism: From: Affine Plane Curve over Rational Field defined by y*s0^3 - 1 To: Affine Plane Curve over Rational Field defined by -x^3 + y^2 Defn: Defined on coordinates by sending (y, s0) to (y*s0, y)))
::
sage: set_verbose(-1) sage: K.<a> = QuadraticField(3) sage: A.<x,y> = AffineSpace(K, 2) sage: C = A.curve(x^4 + 2*x^2 + a*y^3 + 1) sage: C.resolution_of_singularities(extend=True)[0] # long time (2 seconds) (Affine Plane Curve over Number Field in a0 with defining polynomial y^4 - 4*y^2 + 16 defined by 24*x^2*ss1^3 + 24*ss1^3 + (a0^3 - 8*a0), Affine Plane Curve over Number Field in a0 with defining polynomial y^4 - 4*y^2 + 16 defined by 24*s1^2*ss0 + (a0^3 - 8*a0)*ss0^2 + (-6*a0^3)*s1, Affine Plane Curve over Number Field in a0 with defining polynomial y^4 - 4*y^2 + 16 defined by 8*y^2*s0^4 + (4*a0^3)*y*s0^3 - 32*s0^2 + (a0^3 - 8*a0)*y)
::
sage: A.<x,y,z> = AffineSpace(GF(5), 3) sage: C = Curve([y - x^3, (z - 2)^2 - y^3 - x^3], A) sage: R = C.resolution_of_singularities() sage: R[0] (Affine Curve over Finite Field of size 5 defined by x^2 - s1, s1^4 - x*s2^2 + s1, x*s1^3 - s2^2 + x, Affine Curve over Finite Field of size 5 defined by y*s2^2 - y^2 - 1, s2^4 - s0^3 - y^2 - 2, y*s0^3 - s2^2 + y, Affine Curve over Finite Field of size 5 defined by s0^3*s1 + z*s1^3 + s1^4 - 2*s1^3 - 1, z*s0^3 + z*s1^3 - 2*s0^3 - 2*s1^3 - 1, z^2*s1^3 + z*s1^3 - s1^3 - z + s1 + 2)
::
sage: A.<x,y,z,w> = AffineSpace(QQ, 4) sage: C = A.curve([((x - 2)^2 + y^2)^2 - (x - 2)^2 - y^2 + (x - 2)^3, z - y - 7, w - 4]) sage: B = C.resolution_of_singularities() sage: B[0] (Affine Curve over Rational Field defined by s3, s1 - s2, x^2*s2^4 - 4*x*s2^4 + 2*x^2*s2^2 + 4*s2^4 - 8*x*s2^2 + x^2 + 7*s2^2 - 3*x + 1, Affine Curve over Rational Field defined by s3, s2 - 1, y^2*s0^4 + 2*y^2*s0^2 + y*s0^3 + y^2 - s0^2 - 1, Affine Curve over Rational Field defined by s3, s1 - 1, z^2*s0^4 - 14*z*s0^4 + 2*z^2*s0^2 + z*s0^3 + 49*s0^4 - 28*z*s0^2 - 7*s0^3 + z^2 + 97*s0^2 - 14*z + 48, Closed subscheme of Affine Space of dimension 4 over Rational Field defined by: 1)
::
sage: A.<x,y> = AffineSpace(QQ, 2) sage: C = Curve([y - x^2 + 1], A) sage: C.resolution_of_singularities() Traceback (most recent call last): ... TypeError: this curve is already nonsingular
::
sage: A.<x,y> = AffineSpace(QQ, 2) sage: C = A.curve([(x^2 + y^2 - y - 2)*(y - x^2 + 2) + y^3]) sage: C.resolution_of_singularities() Traceback (most recent call last): ... TypeError: this curve has no singular points over its base field. If working over a number field use extend=True """ # helper function for extending the base field (in the case of working over a number field) F = self.base_ring() pts = self.change_ring(F.embeddings(QQbar)[0]).rational_points() L = [t for pt in pts for t in pt] K = number_field_elements_from_algebraics(L)[0] if is_RationalField(K): return F.embeddings(F)[0] else: if is_RationalField(F): return F.embeddings(K)[0] else: # make sure the defining polynomial variable names are the same for K, N N = NumberField(K.defining_polynomial().parent()(F.defining_polynomial()), str(K.gen())) return N.composite_fields(K, both_maps=True)[0][1]*F.embeddings(N)[0] # find the set of singular points of this curve # in the case that the base field is a number field, extend it as needed (if extend == True) raise TypeError("this curve must be irreducible") raise NotImplementedError("this curve must be defined over either a number field or a finite field") C = C.change_ring(extension(C.singular_subscheme())) # the list res holds the data for the patches of the resolution of singularities # each element is a list consisting of the curve defining the patch, a list # of the transition maps from that patch to the other patches, a projection # map from the patch to the original curve, and the set of singular points # of the patch else: " a number field use extend=True") # loop through the patches and blow up each until no patch has singular points # check if there are any singular points in this patch continue # the identity map should be replaced for each of the charts of the blow up # blow up pts[0] # the t-th element of res will be replaced with the new data corresponding to the charts # of the blow up # take out the transition maps from the other resolution patches to the t-th patch # generate the needed data for each patch of the blow up # check if there are any singular points where this patch meets the exceptional divisor # in the case of working over a number field, it might be necessary to extend the base # field in order to find all intersection points emb = extension(X) X = X.change_ring(emb) tmp_curve = B[0][i].change_ring(emb) for pt in X.rational_points(): tmp_pt = tmp_curve([pts[0][i]] + list(pt)) if tmp_curve.is_singular(tmp_pt): n_pts.append(tmp_pt) # avoid needlessly extending the base field if len(n_pts) > 0: # coerce everything to the new base field BC = BC.change_ring(emb) t_maps = [t_maps[j].change_ring(emb) for j in range(len(t_maps))] old_maps = [old_maps[j].change_ring(emb) for j in range(len(old_maps))] pi = pi.change_ring(emb) pts = [pt.change_ring(emb) for pt in pts] # coerce the current blow up data for j in range(len(B[0])): B[0][j] = B[0][j].change_ring(emb) for j in range(len(B[1])): for k in range(len(B[1])): B[1][j][k] = B[1][j][k].change_ring(emb) for j in range(len(B[2])): B[2][j] = B[2][j].change_ring(emb) # coerce the other data in res for j in range(len(res)): res[j][0] = res[j][0].change_ring(emb) for k in range(len(res[j][1])): res[j][1][k] = res[j][1][k].change_ring(emb) res[j][2].change_ring(emb) for k in range(len(res[j][3])): res[j][3][k] = res[j][3][k].change_ring(emb) else: n_pts.append(tmp_pt) # projection map and its inverse j in range(n)] # compose the current transition maps from the original curve to the other patches # with the projection map L[j] = L[j]*t_pi # update transition maps of each other element of res new_t_map = t_pi_inv*old_maps[j] res[j][1].insert(t + i, new_t_map) # create the projection map # singular points # translate the singular points of the parent patch (other than that which was the center of the # blow up) by the inverse of the first projection map # make sure this point is in this chart before attempting to map it try: n_pts.append(t_pi_inv(BC(pts[j]))) except (TypeError, ZeroDivisionError): pass
class AffinePlaneCurve(AffineCurve):
_point = point.AffinePlaneCurvePoint_field
def __init__(self, A, f): r""" Initialization function.
EXAMPLES::
sage: A.<x,y> = AffineSpace(QQ, 2) sage: C = Curve([x^3 - y^2], A); C Affine Plane Curve over Rational Field defined by x^3 - y^2
::
sage: A.<x,y> = AffineSpace(CC, 2) sage: C = Curve([y^2 + x^2], A); C Affine Plane Curve over Complex Field with 53 bits of precision defined by x^2 + y^2 """ raise TypeError("Argument A (= %s) must be an affine plane."%A)
def _repr_type(self): r""" Return a string representation of the type of this curve.
EXAMPLES::
sage: A.<x,y> = AffineSpace(QQ, 2) sage: C = Curve([y - 7/2*x^5 + x - 3], A) sage: C._repr_type() 'Affine Plane' """
def divisor_of_function(self, r): """ Return the divisor of a function on a curve.
INPUT: r is a rational function on X
OUTPUT:
- ``list`` - The divisor of r represented as a list of coefficients and points. (TODO: This will change to a more structural output in the future.)
EXAMPLES::
sage: F = GF(5) sage: P2 = AffineSpace(2, F, names = 'xy') sage: R = P2.coordinate_ring() sage: x, y = R.gens() sage: f = y^2 - x^9 - x sage: C = Curve(f) sage: K = FractionField(R) sage: r = 1/x sage: C.divisor_of_function(r) # todo: not implemented (broken) [[-1, (0, 0, 1)]] sage: r = 1/x^3 sage: C.divisor_of_function(r) # todo: not implemented (broken) [[-3, (0, 0, 1)]] """ F = self.base_ring() f = self.defining_polynomial() pts = self.places_on_curve() numpts = len(pts) R = f.parent() x,y = R.gens() R0 = PolynomialRing(F,3,names = [str(x),str(y),"t"]) vars0 = R0.gens() t = vars0[2] divf = [] for pt0 in pts: if pt0[2] != F(0): lcs = self.local_coordinates(pt0,5) yt = lcs[1] xt = lcs[0] ldg = degree_lowest_rational_function(r(xt,yt),t) if ldg[0] != 0: divf.append([ldg[0],pt0]) return divf
def local_coordinates(self, pt, n): r""" Return local coordinates to precision n at the given point.
Behaviour is flaky - some choices of `n` are worst that others.
INPUT:
- ``pt`` - an F-rational point on X which is not a point of ramification for the projection (x,y) - x.
- ``n`` - the number of terms desired
OUTPUT: x = x0 + t y = y0 + power series in t
EXAMPLES::
sage: F = GF(5) sage: pt = (2,3) sage: R = PolynomialRing(F,2, names = ['x','y']) sage: x,y = R.gens() sage: f = y^2-x^9-x sage: C = Curve(f) sage: C.local_coordinates(pt, 9) [t + 2, -2*t^12 - 2*t^11 + 2*t^9 + t^8 - 2*t^7 - 2*t^6 - 2*t^4 + t^3 - 2*t^2 - 2] """ else:
def plot(self, *args, **kwds): """ Plot the real points on this affine plane curve.
INPUT:
- ``self`` - an affine plane curve
- ``*args`` - optional tuples (variable, minimum, maximum) for plotting dimensions
- ``**kwds`` - optional keyword arguments passed on to ``implicit_plot``
EXAMPLES:
A cuspidal curve::
sage: R.<x, y> = QQ[] sage: C = Curve(x^3 - y^2) sage: C.plot() Graphics object consisting of 1 graphics primitive
A 5-nodal curve of degree 11. This example also illustrates some of the optional arguments::
sage: R.<x, y> = ZZ[] sage: C = Curve(32*x^2 - 2097152*y^11 + 1441792*y^9 - 360448*y^7 + 39424*y^5 - 1760*y^3 + 22*y - 1) sage: C.plot((x, -1, 1), (y, -1, 1), plot_points=400) Graphics object consisting of 1 graphics primitive
A line over `\mathbf{RR}`::
sage: R.<x, y> = RR[] sage: C = Curve(R(y - sqrt(2)*x)) sage: C.plot() Graphics object consisting of 1 graphics primitive """
def is_transverse(self, C, P): r""" Return whether the intersection of this curve with the curve ``C`` at the point ``P`` is transverse.
The intersection at ``P`` is transverse if ``P`` is a nonsingular point of both curves, and if the tangents of the curves at ``P`` are distinct.
INPUT:
- ``C`` -- a curve in the ambient space of this curve.
- ``P`` -- a point in the intersection of both curves.
OUTPUT: Boolean.
EXAMPLES::
sage: A.<x,y> = AffineSpace(QQ, 2) sage: C = Curve([x^2 + y^2 - 1], A) sage: D = Curve([x - 1], A) sage: Q = A([1,0]) sage: C.is_transverse(D, Q) False
::
sage: R.<a> = QQ[] sage: K.<b> = NumberField(a^3 + 2) sage: A.<x,y> = AffineSpace(K, 2) sage: C = A.curve([x*y]) sage: D = A.curve([y - b*x]) sage: Q = A([0,0]) sage: C.is_transverse(D, Q) False
::
sage: A.<x,y> = AffineSpace(QQ, 2) sage: C = Curve([y - x^3], A) sage: D = Curve([y + x], A) sage: Q = A([0,0]) sage: C.is_transverse(D, Q) True """ raise TypeError("(=%s) must be a point in the intersection of (=%s) and this curve"%(P,C))
# there is only one tangent at a nonsingular point of a plane curve
def multiplicity(self, P): r""" Return the multiplicity of this affine plane curve at the point ``P``.
In the special case of affine plane curves, the multiplicity of an affine plane curve at the point (0,0) can be computed as the minimum of the degrees of the homogeneous components of its defining polynomial. To compute the multiplicity of a different point, a linear change of coordinates is used.
This curve must be defined over a field. An error if raised if ``P`` is not a point on this curve.
INPUT:
- ``P`` -- a point in the ambient space of this curve.
OUTPUT:
An integer.
EXAMPLES::
sage: A.<x,y> = AffineSpace(QQ, 2) sage: C = Curve([y^2 - x^3], A) sage: Q1 = A([1,1]) sage: C.multiplicity(Q1) 1 sage: Q2 = A([0,0]) sage: C.multiplicity(Q2) 2
::
sage: A.<x,y> = AffineSpace(QQbar,2) sage: C = Curve([-x^7 + (-7)*x^6 + y^6 + (-21)*x^5 + 12*y^5 + (-35)*x^4 + 60*y^4 +\ (-35)*x^3 + 160*y^3 + (-21)*x^2 + 240*y^2 + (-7)*x + 192*y + 63], A) sage: Q = A([-1,-2]) sage: C.multiplicity(Q) 6
::
sage: A.<x,y> = AffineSpace(QQ, 2) sage: C = A.curve([y^3 - x^3 + x^6]) sage: Q = A([1,1]) sage: C.multiplicity(Q) Traceback (most recent call last): ... TypeError: (=(1, 1)) is not a point on (=Affine Plane Curve over Rational Field defined by x^6 - x^3 + y^3) """ raise TypeError("curve must be defined over a field")
# Check whether P is a point on this curve
# Apply a linear change of coordinates to self so that P becomes (0,0)
# Compute the multiplicity of the new curve at (0,0), which is the minimum of the degrees of its # nonzero terms
def tangents(self, P, factor=True): r""" Return the tangents of this affine plane curve at the point ``P``.
The point ``P`` must be a point on this curve.
INPUT:
- ``P`` -- a point on this curve.
- ``factor`` -- (default: True) whether to attempt computing the polynomials of the individual tangent lines over the base field of this curve, or to just return the polynomial corresponding to the union of the tangent lines (which requires fewer computations).
OUTPUT:
- a list of polynomials in the coordinate ring of the ambient space of this curve.
EXAMPLES::
sage: set_verbose(-1) sage: A.<x,y> = AffineSpace(QQbar, 2) sage: C = Curve([x^5*y^3 + 2*x^4*y^4 + x^3*y^5 + 3*x^4*y^3 + 6*x^3*y^4 + 3*x^2*y^5\ + 3*x^3*y^3 + 6*x^2*y^4 + 3*x*y^5 + x^5 + 10*x^4*y + 40*x^3*y^2 + 81*x^2*y^3 + 82*x*y^4\ + 33*y^5], A) sage: Q = A([0,0]) sage: C.tangents(Q) [x + 3.425299577684700?*y, x + (1.949159013086856? + 1.179307909383728?*I)*y, x + (1.949159013086856? - 1.179307909383728?*I)*y, x + (1.338191198070795? + 0.2560234251008043?*I)*y, x + (1.338191198070795? - 0.2560234251008043?*I)*y] sage: C.tangents(Q, factor=False) [120*x^5 + 1200*x^4*y + 4800*x^3*y^2 + 9720*x^2*y^3 + 9840*x*y^4 + 3960*y^5]
::
sage: R.<a> = QQ[] sage: K.<b> = NumberField(a^2 - 3) sage: A.<x,y> = AffineSpace(K, 2) sage: C = Curve([(x^2 + y^2 - 2*x)^2 - x^2 - y^2], A) sage: Q = A([0,0]) sage: C.tangents(Q) [x + (-1/3*b)*y, x + (1/3*b)*y]
::
sage: A.<x,y> = AffineSpace(QQ, 2) sage: C = A.curve([y^2 - x^3 - x^2]) sage: Q = A([0,0]) sage: C.tangents(Q) [x - y, x + y]
::
sage: A.<x,y> = AffineSpace(QQ, 2) sage: C = A.curve([y*x - x^4 + 2*x^2]) sage: Q = A([1,1]) sage: C.tangents(Q) Traceback (most recent call last): ... TypeError: (=(1, 1)) is not a point on (=Affine Plane Curve over Rational Field defined by -x^4 + 2*x^2 + x*y) """ # move P to (0,0) # first add tangents corresponding to vars[0], vars[1] if they divide T # vars[0] divides T fact.append(vars[0]) # divide T by that power of vars[0] T = self.ambient_space().coordinate_ring()(dict([((v[0] - t,v[1]), h) for (v,h) in T.dict().items()])) # vars[1] divides T fact.append(vars[1]) # divide T by that power of vars[1] T = self.ambient_space().coordinate_ring()(dict([((v[0],v[1] - t), h) for (v,h) in T.dict().items()])) # T is homogeneous in var[0], var[1] if nonconstant, so dehomogenize else: T = T(1, vars[1]) roots = T.univariate_polynomial().roots() fact.extend([vars[1] - roots[i][0]*vars[0] for i in range(len(roots))]) else:
def is_ordinary_singularity(self, P): r""" Return whether the singular point ``P`` of this affine plane curve is an ordinary singularity.
The point ``P`` is an ordinary singularity of this curve if it is a singular point, and if the tangents of this curve at ``P`` are distinct.
INPUT:
- ``P`` -- a point on this curve.
OUTPUT:
- Boolean. True or False depending on whether ``P`` is or is not an ordinary singularity of this curve, respectively. An error is raised if ``P`` is not a singular point of this curve.
EXAMPLES::
sage: A.<x,y> = AffineSpace(QQ, 2) sage: C = Curve([y^2 - x^3], A) sage: Q = A([0,0]) sage: C.is_ordinary_singularity(Q) False
::
sage: R.<a> = QQ[] sage: K.<b> = NumberField(a^2 - 3) sage: A.<x,y> = AffineSpace(K, 2) sage: C = Curve([(x^2 + y^2 - 2*x)^2 - x^2 - y^2], A) sage: Q = A([0,0]) sage: C.is_ordinary_singularity(Q) True
::
sage: A.<x,y> = AffineSpace(QQ, 2) sage: C = A.curve([x^2*y - y^2*x + y^2 + x^3]) sage: Q = A([-1,-1]) sage: C.is_ordinary_singularity(Q) Traceback (most recent call last): ... TypeError: (=(-1, -1)) is not a singular point of (=Affine Plane Curve over Rational Field defined by x^3 + x^2*y - x*y^2 + y^2) """
# use resultants to determine if there is a higher multiplicity tangent else:
def rational_parameterization(self): r""" Return a rational parameterization of this curve.
This curve must have rational coefficients and be absolutely irreducible (i.e. irreducible over the algebraic closure of the rational field). The curve must also be rational (have geometric genus zero).
The rational parameterization may have coefficients in a quadratic extension of the rational field.
OUTPUT:
- a birational map between `\mathbb{A}^{1}` and this curve, given as a scheme morphism.
EXAMPLES::
sage: A.<x,y> = AffineSpace(QQ, 2) sage: C = Curve([y^2 - x], A) sage: C.rational_parameterization() Scheme morphism: From: Affine Space of dimension 1 over Rational Field To: Affine Plane Curve over Rational Field defined by y^2 - x Defn: Defined on coordinates by sending (t) to (t^2, t)
::
sage: A.<x,y> = AffineSpace(QQ, 2) sage: C = Curve([(x^2 + y^2 - 2*x)^2 - x^2 - y^2], A) sage: C.rational_parameterization() Scheme morphism: From: Affine Space of dimension 1 over Rational Field To: Affine Plane Curve over Rational Field defined by x^4 + 2*x^2*y^2 + y^4 - 4*x^3 - 4*x*y^2 + 3*x^2 - y^2 Defn: Defined on coordinates by sending (t) to ((-12*t^4 + 6*t^3 + 4*t^2 - 2*t)/(-25*t^4 + 40*t^3 - 26*t^2 + 8*t - 1), (-9*t^4 + 12*t^3 - 4*t + 1)/(-25*t^4 + 40*t^3 - 26*t^2 + 8*t - 1))
::
sage: A.<x,y> = AffineSpace(QQ, 2) sage: C = Curve([x^2 + y^2 + 7], A) sage: C.rational_parameterization() Scheme morphism: From: Affine Space of dimension 1 over Number Field in a with defining polynomial a^2 + 7 To: Affine Plane Curve over Number Field in a with defining polynomial a^2 + 7 defined by x^2 + y^2 + 7 Defn: Defined on coordinates by sending (t) to ((-7*t^2 + 7)/((-a)*t^2 + (-a)), 14*t/((-a)*t^2 + (-a))) """ # these polynomials are homogeneous in two indeterminants, so dehomogenize wrt one of the variables # because of the parameter i=0, the projective closure is constructed with respect to the # affine patch corresponding to the first coordinate being nonzero. Thus para[0] will not be # the zero polynomial, and dehomogenization won't change this
def fundamental_group(self): r""" Return a presentation of the fundamental group of the complement of ``self``.
.. NOTE::
The curve must be defined over the rationals or a number field with an embedding over `\QQbar`.
EXAMPLES::
sage: A.<x,y> = AffineSpace(QQ, 2) sage: C = A.curve(y^2 - x^3 - x^2) sage: C.fundamental_group() # optional - sirocco Finitely presented group < x0 | >
In the case of number fields, they need to have an embedding to the algebraic field::
sage: a = QQ[x](x^2+5).roots(QQbar)[0][0] sage: F = NumberField(a.minpoly(), 'a', embedding=a) sage: F.inject_variables() Defining a sage: A.<x,y> = AffineSpace(F, 2) sage: C = A.curve(y^2 - a*x^3 - x^2) sage: C.fundamental_group() # optional - sirocco Finitely presented group < x0 | >
.. WARNING::
This functionality requires the sirocco package to be installed. """ from sage.schemes.curves.zariski_vankampen import fundamental_group F = self.base_ring() from sage.rings.qqbar import QQbar if QQbar.coerce_map_from(F) is None: raise NotImplementedError("the base field must have an embedding" " to the algebraic field") f = self.defining_polynomial() return fundamental_group(f, projective=False)
def riemann_surface(self,**kwargs): r"""Return the complex riemann surface determined by this curve
OUTPUT:
- RiemannSurface object
EXAMPLES::
sage: R.<x,y>=QQ[] sage: C=Curve(x^3+3*y^3+5) sage: C.riemann_surface() Riemann surface defined by polynomial f = x^3 + 3*y^3 + 5 = 0, with 53 bits of precision
"""
class AffinePlaneCurve_finite_field(AffinePlaneCurve):
_point = point.AffinePlaneCurvePoint_finite_field
def rational_points(self, algorithm="enum"): r""" Return sorted list of all rational points on this curve.
Use *very* naive point enumeration to find all rational points on this curve over a finite field.
EXAMPLES::
sage: A.<x,y> = AffineSpace(2,GF(9,'a')) sage: C = Curve(x^2 + y^2 - 1) sage: C Affine Plane Curve over Finite Field in a of size 3^2 defined by x^2 + y^2 - 1 sage: C.rational_points() [(0, 1), (0, 2), (1, 0), (2, 0), (a + 1, a + 1), (a + 1, 2*a + 2), (2*a + 2, a + 1), (2*a + 2, 2*a + 2)] """
class AffinePlaneCurve_prime_finite_field(AffinePlaneCurve_finite_field): # CHECK WHAT ASSUMPTIONS ARE MADE REGARDING AFFINE VS. PROJECTIVE MODELS!!! # THIS IS VERY DIRTY STILL -- NO DATASTRUCTURES FOR DIVISORS.
def riemann_roch_basis(self,D): """ Interfaces with Singular's BrillNoether command.
INPUT:
- ``self`` - a plane curve defined by a polynomial eqn f(x,y) = 0 over a prime finite field F = GF(p) in 2 variables x,y representing a curve X: f(x,y) = 0 having n F-rational points (see the Sage function places_on_curve)
- ``D`` - an n-tuple of integers `(d1, ..., dn)` representing the divisor `Div = d1*P1+...+dn*Pn`, where `X(F) = \{P1,...,Pn\}`. *The ordering is that dictated by places_on_curve.*
OUTPUT: basis of L(Div)
EXAMPLES::
sage: R = PolynomialRing(GF(5),2,names = ["x","y"]) sage: x, y = R.gens() sage: f = y^2 - x^9 - x sage: C = Curve(f) sage: D = [6,0,0,0,0,0] sage: C.riemann_roch_basis(D) [1, (y^2*z^4 - x*z^5)/x^6, (y^2*z^5 - x*z^6)/x^7, (y^2*z^6 - x*z^7)/x^8] """
def rational_points(self, algorithm="enum"): r""" Return sorted list of all rational points on this curve.
INPUT:
- ``algorithm`` - string:
+ ``'enum'`` - straightforward enumeration
+ ``'bn'`` - via Singular's Brill-Noether package.
+ ``'all'`` - use all implemented algorithms and verify that they give the same answer, then return it
.. note::
The Brill-Noether package does not always work. When it fails a RuntimeError exception is raised.
EXAMPLES::
sage: x, y = (GF(5)['x,y']).gens() sage: f = y^2 - x^9 - x sage: C = Curve(f); C Affine Plane Curve over Finite Field of size 5 defined by -x^9 + y^2 - x sage: C.rational_points(algorithm='bn') [(0, 0), (2, 2), (2, 3), (3, 1), (3, 4)] sage: C = Curve(x - y + 1) sage: C.rational_points() [(0, 1), (1, 2), (2, 3), (3, 4), (4, 0)]
We compare Brill-Noether and enumeration::
sage: x, y = (GF(17)['x,y']).gens() sage: C = Curve(x^2 + y^5 + x*y - 19) sage: v = C.rational_points(algorithm='bn') sage: w = C.rational_points(algorithm='enum') sage: len(v) 20 sage: v == w True """
except (TypeError, RuntimeError) as s: raise RuntimeError(str(s) + "\n\n ** Unable to use the Brill-Noether Singular package to compute all points (see above).")
# We use sage_flattened_str_list since iterating through # the entire list through the sage/singular interface directly # would involve hundreds of calls to singular, and timing issues # with the expect interface could crop up. Also, this is vastly # faster (and more robust). # remove multiple points
elif algorithm == "all":
S_enum = self.rational_points(algorithm = "enum") S_bn = self.rational_points(algorithm = "bn") if S_enum != S_bn: raise RuntimeError("Bug in rational_points -- different algorithms give different answers for curve %s!"%self) return S_enum
else: raise ValueError("No algorithm '%s' known"%algorithm) |