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""" Curve points
EXAMPLES:
We can create points on projective curves::
sage: P.<x,y,z,w> = ProjectiveSpace(QQ, 3) sage: C = Curve([x^3 - 2*x*z^2 - y^3, z^3 - w^3 - x*y*z], P) sage: Q = C([1,1,0,0]) sage: type(Q) <class 'sage.schemes.curves.point.ProjectiveCurvePoint_field'> sage: Q.parent() Set of rational points of Projective Curve over Rational Field defined by x^3 - y^3 - 2*x*z^2, -x*y*z + z^3 - w^3
or on affine curves::
sage: A.<x,y> = AffineSpace(GF(23), 2) sage: C = Curve([y - y^4 + 17*x^2 - 2*x + 22], A) sage: Q = C([22,21]) sage: type(Q) <class 'sage.schemes.curves.point.AffinePlaneCurvePoint_finite_field'> sage: Q.parent() Set of rational points of Affine Plane Curve over Finite Field of size 23 defined by -y^4 - 6*x^2 - 2*x + y - 1
AUTHORS:
- Grayson Jorgenson (2016-6): initial version """
#***************************************************************************** # 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 sage.schemes.affine.affine_point import (SchemeMorphism_point_affine_field, SchemeMorphism_point_affine_finite_field) from sage.schemes.projective.projective_point import (SchemeMorphism_point_projective_field, SchemeMorphism_point_projective_finite_field)
class ProjectiveCurvePoint_field(SchemeMorphism_point_projective_field):
def is_singular(self): r""" Return whether this point is a singular point of the projective curve it is on.
OUTPUT: Boolean.
EXAMPLES::
sage: P.<x,y,z,w> = ProjectiveSpace(QQ, 3) sage: C = Curve([x^2 - y^2, z - w], P) sage: Q1 = C([0,0,1,1]) sage: Q1.is_singular() True sage: Q2 = C([1,1,1,1]) sage: Q2.is_singular() False """
class ProjectivePlaneCurvePoint_field(ProjectiveCurvePoint_field):
def multiplicity(self): r""" Return the multiplicity of this point with respect to the projective curve it is on.
OUTPUT: Integer.
EXAMPLES::
sage: P.<x,y,z> = ProjectiveSpace(GF(17), 2) sage: C = Curve([y^3*z - 16*x^4], P) sage: Q = C([0,0,1]) sage: Q.multiplicity() 3 """
def tangents(self): r""" Return the tangents at this point of the projective plane curve this point is on.
OUTPUT:
- a list of polynomials in the coordinate ring of the ambient space of the curve this point is on.
EXAMPLES::
sage: P.<x,y,z> = ProjectiveSpace(QQ, 2) sage: C = Curve([y^2*z^3 - x^5 + 18*y*x*z^3]) sage: Q = C([0,0,1]) sage: Q.tangents() [y, 18*x + y] """
def is_ordinary_singularity(self): r""" Return whether this point is an ordinary singularity of the projective plane curve it is on.
OUTPUT: Boolean.
EXAMPLES::
sage: P.<x,y,z> = ProjectiveSpace(QQ, 2) sage: C = Curve([z^6 - x^6 - x^3*z^3 - x^3*y^3]) sage: Q = C([0,1,0]) sage: Q.is_ordinary_singularity() False
::
sage: R.<a> = QQ[] sage: K.<b> = NumberField(a^2 - 3) sage: P.<x,y,z> = ProjectiveSpace(K, 2) sage: C = P.curve([x^2*y^3*z^4 - y^6*z^3 - 4*x^2*y^4*z^3 - 4*x^4*y^2*z^3 + 3*y^7*z^2 + 10*x^2*y^5*z^2\ + 9*x^4*y^3*z^2 + 5*x^6*y*z^2 - 3*y^8*z - 9*x^2*y^6*z - 11*x^4*y^4*z - 7*x^6*y^2*z - 2*x^8*z + y^9\ + 2*x^2*y^7 + 3*x^4*y^5 + 4*x^6*y^3 + 2*x^8*y]) sage: Q = C([-1/2, 1/2, 1]) sage: Q.is_ordinary_singularity() True """
def is_transverse(self, D): r""" Return whether the intersection of the curve ``D`` at this point with the curve this point is on is transverse or not.
INPUT:
- ``D`` -- a curve in the same ambient space as the curve this point is on.
OUTPUT: Boolean.
EXAMPLES::
sage: P.<x,y,z> = ProjectiveSpace(QQ, 2) sage: C = Curve([x^2 - 2*y^2 - 2*z^2], P) sage: D = Curve([y - z], P) sage: Q = C([2,1,1]) sage: Q.is_transverse(D) True
::
sage: P.<x,y,z> = ProjectiveSpace(GF(17), 2) sage: C = Curve([x^4 - 16*y^3*z], P) sage: D = Curve([y^2 - z*x], P) sage: Q = C([0,0,1]) sage: Q.is_transverse(D) False """
class ProjectivePlaneCurvePoint_finite_field(ProjectivePlaneCurvePoint_field, SchemeMorphism_point_projective_finite_field): pass
class AffineCurvePoint_field(SchemeMorphism_point_affine_field):
def is_singular(self): r""" Return whether this point is a singular point of the affine curve it is on.
OUTPUT: Boolean.
EXAMPLES::
sage: K = QuadraticField(-1) sage: A.<x,y,z> = AffineSpace(K, 3) sage: C = Curve([(x^4 + 2*z + 2)*y, z - y + 1]) sage: Q1 = C([0,0,-1]) sage: Q1.is_singular() True sage: Q2 = C([-K.gen(),0,-1]) sage: Q2.is_singular() False """
class AffinePlaneCurvePoint_field(AffineCurvePoint_field):
def multiplicity(self): r""" Return the multiplicity of this point with respect to the affine curve it is on.
OUTPUT: Integer.
EXAMPLES::
sage: A.<x,y> = AffineSpace(QQ, 2) sage: C = A.curve([2*x^7 - 3*x^6*y + x^5*y^2 + 31*x^6 - 40*x^5*y + 13*x^4*y^2 - x^3*y^3\ + 207*x^5 - 228*x^4*y + 70*x^3*y^2 - 7*x^2*y^3 + 775*x^4 - 713*x^3*y + 193*x^2*y^2 - 19*x*y^3\ + y^4 + 1764*x^3 - 1293*x^2*y + 277*x*y^2 - 22*y^3 + 2451*x^2 - 1297*x*y + 172*y^2 + 1935*x\ - 570*y + 675]) sage: Q = C([-2,1]) sage: Q.multiplicity() 4 """
def tangents(self): r""" Return the tangents at this point of the affine plane curve this point is on.
OUTPUT:
- a list of polynomials in the coordinate ring of the ambient space of the curve this point is on.
EXAMPLES::
sage: A.<x,y> = AffineSpace(QQ, 2) sage: C = A.curve([x^5 - x^3*y^2 + 5*x^4 - x^3*y - 3*x^2*y^2 + x*y^3 + 10*x^3 - 3*x^2*y -\ 3*x*y^2 + y^3 + 10*x^2 - 3*x*y - y^2 + 5*x - y + 1]) sage: Q = C([-1,0]) sage: Q.tangents() [y, x + 1, x - y + 1, x + y + 1] """
def is_ordinary_singularity(self): r""" Return whether this point is an ordinary singularity of the affine plane curve it is on.
OUTPUT: Boolean.
EXAMPLES::
sage: A.<x,y> = AffineSpace(QQ, 2) sage: C = A.curve([x^5 - x^3*y^2 + 5*x^4 - x^3*y - 3*x^2*y^2 + x*y^3 + 10*x^3 - 3*x^2*y -\ 3*x*y^2 + y^3 + 10*x^2 - 3*x*y - y^2 + 5*x - y + 1]) sage: Q = C([-1,0]) sage: Q.is_ordinary_singularity() True
::
sage: A.<x,y> = AffineSpace(GF(7), 2) sage: C = A.curve([y^2 - x^7 - 6*x^3]) sage: Q = C([0,0]) sage: Q.is_ordinary_singularity() False """
def is_transverse(self, D): r""" Return whether the intersection of the curve ``D`` at this point with the curve this point is on is transverse or not.
INPUT:
- ``D`` -- a curve in the same ambient space as the curve this point is on.
OUTPUT: Boolean.
EXAMPLES::
sage: A.<x,y> = AffineSpace(QQ, 2) sage: C = Curve([y - x^2], A) sage: D = Curve([y], A) sage: Q = C([0,0]) sage: Q.is_transverse(D) False
::
sage: R.<a> = QQ[] sage: K.<b> = NumberField(a^2 - 2) sage: A.<x,y> = AffineSpace(K, 2) sage: C = Curve([y^2 + x^2 - 1], A) sage: D = Curve([y - x], A) sage: Q = C([-1/2*b,-1/2*b]) sage: Q.is_transverse(D) True """
class AffinePlaneCurvePoint_finite_field(AffinePlaneCurvePoint_field, SchemeMorphism_point_affine_finite_field): pass |