Rational points of curves

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: 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: 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

class sage.schemes.curves.point.AffineCurvePoint_field(X, v, check=True)

Bases: sage.schemes.affine.affine_point.SchemeMorphism_point_affine_field

is_singular()

Return whether this point is a singular point of the affine curve it is on.

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 sage.schemes.curves.point.AffinePlaneCurvePoint_field(X, v, check=True)

Bases: sage.schemes.curves.point.AffineCurvePoint_field

Point of an affine plane curve over a field.

is_ordinary_singularity()

Return whether this point is an ordinary singularity of the affine plane curve it 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.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

is_transverse(D)

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.

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

multiplicity()

Return the multiplicity of this point with respect to the affine curve it is on.

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

tangents()

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]

class sage.schemes.curves.point.AffinePlaneCurvePoint_finite_field(X, v, check=True)

Bases: sage.schemes.curves.point.AffinePlaneCurvePoint_field, sage.schemes.affine.affine_point.SchemeMorphism_point_affine_finite_field

Point of an affine plane curve over a finite field.

class sage.schemes.curves.point.IntegralAffineCurvePoint(X, v, check=True)

Bases: sage.schemes.curves.point.AffineCurvePoint_field

Point of an integral affine curve.

closed_point()

Return the closed point that corresponds to this rational point.

EXAMPLES:

sage: A.<x,y> = AffineSpace(GF(8), 2)
sage: C = Curve(x^5 + y^5 + x*y + 1)
sage: p = C([1,1])
sage: p.closed_point()
Point (x + 1, y + 1)

place()

Return a place on this point.

EXAMPLES:

sage: A.<x,y> = AffineSpace(GF(2), 2)
sage: C = Curve(x^5 + y^5 + x*y + 1)
sage: p = C(-1,-1)
sage: p
(1, 1)
sage: p.closed_point()
Point (x + 1, y + 1)
sage: _.place()
Place (x + 1, (1/(x^5 + 1))*y^4 + ((x^5 + x^4 + 1)/(x^5 + 1))*y^3 +
((x^5 + x^3 + 1)/(x^5 + 1))*y^2 + (x^2/(x^5 + 1))*y)

places()

Return all places on this point.

EXAMPLES:

sage: A.<x,y> = AffineSpace(GF(2), 2)
sage: C = Curve(x^5 + y^5 + x*y + 1)
sage: p = C(-1,-1)
sage: p
(1, 1)
sage: p.closed_point()
Point (x + 1, y + 1)
sage: _.places()
[Place (x + 1, (1/(x^5 + 1))*y^4 + ((x^5 + x^4 + 1)/(x^5 + 1))*y^3
+ ((x^5 + x^3 + 1)/(x^5 + 1))*y^2 + (x^2/(x^5 + 1))*y), Place (x +
1, (1/(x^5 + 1))*y^4 + ((x^5 + x^4 + 1)/(x^5 + 1))*y^3 + (x^3/(x^5
+ 1))*y^2 + (x^2/(x^5 + 1))*y + x + 1)]

class sage.schemes.curves.point.IntegralAffineCurvePoint_finite_field(X, v, check=True)

Bases: sage.schemes.curves.point.IntegralAffineCurvePoint

Point of an integral affine curve over a finite field.

class sage.schemes.curves.point.IntegralAffinePlaneCurvePoint(X, v, check=True)

Bases: sage.schemes.curves.point.IntegralAffineCurvePoint, sage.schemes.curves.point.AffinePlaneCurvePoint_field

Point of an integral affine plane curve over a finite field.

class sage.schemes.curves.point.IntegralAffinePlaneCurvePoint_finite_field(X, v, check=True)

Bases: sage.schemes.curves.point.AffinePlaneCurvePoint_finite_field, sage.schemes.curves.point.IntegralAffineCurvePoint_finite_field

Point of an integral affine plane curve over a finite field.

class sage.schemes.curves.point.IntegralProjectiveCurvePoint(X, v, check=True)

Bases: sage.schemes.curves.point.ProjectiveCurvePoint_field

closed_point()

Return the closed point corresponding to this rational point.

EXAMPLES:

sage: P.<x,y,z> = ProjectiveSpace(GF(17), 2)
sage: C = Curve([x^4 - 16*y^3*z], P)
sage: C.singular_points()
[(0 : 0 : 1)]
sage: p = _[0]
sage: p.closed_point()
Point (x, y)

place()

Return a place on this point.

EXAMPLES:

sage: P.<x,y,z> = ProjectiveSpace(GF(17), 2)
sage: C = Curve([x^4 - 16*y^3*z], P)
sage: C.singular_points()
[(0 : 0 : 1)]
sage: p = _[0]
sage: p.place()
Place (y)

places()

Return all places on this point.

EXAMPLES:

sage: P.<x,y,z> = ProjectiveSpace(GF(17), 2)
sage: C = Curve([x^4 - 16*y^3*z], P)
sage: C.singular_points()
[(0 : 0 : 1)]
sage: p = _[0]
sage: p.places()
[Place (y)]

class sage.schemes.curves.point.IntegralProjectiveCurvePoint_finite_field(X, v, check=True)

Bases: sage.schemes.curves.point.IntegralProjectiveCurvePoint

Point of an integral projective curve over a finite field.

class sage.schemes.curves.point.IntegralProjectivePlaneCurvePoint(X, v, check=True)

Bases: sage.schemes.curves.point.IntegralProjectiveCurvePoint, sage.schemes.curves.point.ProjectivePlaneCurvePoint_field

Point of an integral projective plane curve over a field.

class sage.schemes.curves.point.IntegralProjectivePlaneCurvePoint_finite_field(X, v, check=True)

Bases: sage.schemes.curves.point.ProjectivePlaneCurvePoint_finite_field, sage.schemes.curves.point.IntegralProjectiveCurvePoint_finite_field

Point of an integral projective plane curve over a finite field.

class sage.schemes.curves.point.ProjectiveCurvePoint_field(X, v, check=True)

Bases: sage.schemes.projective.projective_point.SchemeMorphism_point_projective_field

Point of a projective curve over a field.

is_singular()

Return whether this point is a singular point of the projective curve it is on.

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 sage.schemes.curves.point.ProjectivePlaneCurvePoint_field(X, v, check=True)

Bases: sage.schemes.curves.point.ProjectiveCurvePoint_field

Point of a projective plane curve over a field.

is_ordinary_singularity()

Return whether this point is an ordinary singularity of the projective plane curve it is on.

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

is_transverse(D)

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

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

multiplicity()

Return the multiplicity of this point with respect to the projective curve it is on.

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

tangents()

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]

class sage.schemes.curves.point.ProjectivePlaneCurvePoint_finite_field(X, v, check=True)

Bases: sage.schemes.curves.point.ProjectivePlaneCurvePoint_field, sage.schemes.projective.projective_point.SchemeMorphism_point_projective_finite_field

Point of a projective plane curve over a finite field.