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20774: Changes from first review.
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Have not yet addressed class structure to accomodate curves
defined specifically over fields, and have not
implemented corresponding singularity analysis functionality
for points on curves yet.
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@gsjorgenson
gsjorgenson committed Jun 11, 2016
1 parent e1f1529 commit e6f1a93
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77 changes: 48 additions & 29 deletions src/sage/schemes/curves/affine_curve.py
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Original file line number Diff line number Diff line change
Expand Up @@ -149,52 +149,56 @@ def multiplicity(self, P):
Return the multiplicity of this affine curve at the point ``P``.
This is computed as the multiplicity of the local ring of self corresponding to ``P``. This
curve must be defined over a field.
curve must be defined over a field. An error is raised if ``P`` is not a point on this curve.
INPUT:
- ``P`` -- a point in the ambient space of this curve.
OUTPUT:
- an integer.
An integer.
EXAMPLES::
sage: A.<x,y,z> = AffineSpace(QQ, 3)
sage: A.<x,y,z> = AffineSpace(CC, 3)
sage: C = A.curve([y - x^2, z - x^3])
sage: Q1 = A([1,1,1])
sage: C.multiplicity(Q1)
sage: Q = A([1,1,1])
sage: C.multiplicity(Q)
1
sage: Q2 = A([1,2,1])
sage: C.multiplicity(Q2)
0
::
sage: A.<x,y,z,w> = AffineSpace(QQ, 4)
sage: C = A.curve([y^9 - x^5, z^10 - w - y^4, z - y])
sage: C.multiplicity(A([0,0,0,0]))
5
::
sage: A.<x,y,z,w,v> = AffineSpace(GF(23), 5)
sage: C = A.curve([x^8 - y, y^7 - z, z^3 - 1, w^5 - v^3])
sage: Q = A([22,1,1,0,0])
sage: C.multiplicity(Q)
3
"""
if not self.base_ring() in Fields():
raise TypeError("curve must be defined over a field")

# Check whether P is in the ambient space of this curve
# Check whether P is a point on this curve
try:
P = self.ambient_space()(P)
P = self(P)
except TypeError:
raise TypeError("(=%s) must be a point on (=%s)"%(P,self))

# Apply a linear change of coordinates to self so that P is sent to the origin
# and then compute the multiplicity of the local ring of the translated curve
# corresponding to the point (0,...,0)
AA = self.ambient_space()
chng_coords = [AA.gens()[i] + P[i] for i in range(AA.dimension_relative())]
I = AA.coordinate_ring().ideal([f(chng_coords) for f in self.defining_polynomials()])

# Compute the multiplicity of the local ring of the new curve defined by I
# corresponding to the point (0,...,0)
R = AA.coordinate_ring().change_ring(order='negdegrevlex')
return singular.mult(singular.std(I.change_ring(R))).sage()
I = R.ideal([f(chng_coords) for f in self.defining_polynomials()])
return singular.mult(singular.std(I)).sage()

class AffinePlaneCurve(AffineCurve):
def __init__(self, A, f):
Expand Down Expand Up @@ -411,38 +415,45 @@ def multiplicity(self, P):
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.
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.
An integer.
EXAMPLES::
sage: A.<x,y> = AffineSpace(QQ, 2)
sage: C = Curve([y^2 - x^3], A)
sage: Q1 = A([1,0])
sage: Q1 = A([1,1])
sage: C.multiplicity(Q1)
0
sage: Q2 = A([1,1])
sage: C.multiplicity(Q2)
1
sage: Q3 = A([0,0])
sage: C.multiplicity(Q3)
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
"""
if not self.base_ring() in Fields():
raise TypeError("curve must be defined over a field")

# Check whether P in in the ambient space of this curve
# Check whether P is a point on this curve
try:
P = self.ambient_space()(P)
P = self(P)
except TypeError:
raise TypeError("(=%s) must be a point in the ambient space of (=%s)"%(P,self))
raise TypeError("(=%s) must be a point on (=%s)"%(P,self))

# Apply a linear change of coordinates to self so that P becomes (0,0)
AA = self.ambient_space()
Expand All @@ -456,6 +467,8 @@ def tangents(self, P):
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.
Expand All @@ -473,10 +486,16 @@ def tangents(self, P):
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]
"""
r = self.multiplicity(P)
if r < 1:
raise TypeError("(=%s) must be a point on (=%s)"%(P,self))
f = self.defining_polynomials()[0]
vars = self.ambient_space().gens()
deriv = [f.derivative(vars[0],i).derivative(vars[1],r-i)(list(P)) for i in range(r+1)]
Expand All @@ -499,7 +518,7 @@ def is_ordinary_singularity(self, P):
OUTPUT:
- Boolean. True or False depending on whether ``P`` is or is not an ordinary singularity of this
curve, respectively.
curve, respectively. An error is raised if ``P`` is not a singular point of this curve.
EXAMPLES::
Expand Down
76 changes: 74 additions & 2 deletions src/sage/schemes/curves/curve.py
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Original file line number Diff line number Diff line change
Expand Up @@ -197,6 +197,44 @@ def union(self, other):

__add__ = union

def singular_subscheme(self):
r"""
Return the subscheme of singular points of this curve.
OUTPUT:
- a subscheme in the ambient space of this curve.
EXAMPLES::
sage: A.<x,y> = AffineSpace(CC, 2)
sage: C = Curve([y^4 - 2*x^5 - x^2*y], A)
sage: C.singular_subscheme()
Closed subscheme of Affine Space of dimension 2 over Complex Field with
53 bits of precision defined by:
(-2.00000000000000)*x^5 + y^4 - x^2*y,
(-10.0000000000000)*x^4 + (-2.00000000000000)*x*y,
4.00000000000000*y^3 - x^2
::
sage: P.<x,y,z,w> = ProjectiveSpace(QQ, 3)
sage: C = Curve([y^8 - x^2*z*w^5, w^2 - 2*y^2 - x*z], P)
sage: C.singular_subscheme()
Closed subscheme of Projective Space of dimension 3 over Rational Field
defined by:
y^8 - x^2*z*w^5,
-2*y^2 - x*z + w^2,
-x^3*y*z^4 + 3*x^2*y*z^3*w^2 - 3*x*y*z^2*w^4 + 8*x*y*z*w^5 + y*z*w^6,
x^2*z*w^5,
-5*x^2*z^2*w^4 - 4*x*z*w^6,
x^4*y*z^3 - 3*x^3*y*z^2*w^2 + 3*x^2*y*z*w^4 - 4*x^2*y*w^5 - x*y*w^6,
-2*x^3*y*z^3*w + 6*x^2*y*z^2*w^3 - 20*x^2*y*z*w^4 - 6*x*y*z*w^5 +
2*y*w^7,
-5*x^3*z*w^4 - 2*x^2*w^6
"""
return self.ambient_space().subscheme(self.Jacobian())

def singular_points(self, F=None):
r"""
Return the set of singular points of this curve.
Expand Down Expand Up @@ -235,5 +273,39 @@ def singular_points(self, F=None):
raise TypeError("curve must be defined over a field")
elif not F in Fields():
raise TypeError("(=%s) must be a field"%F)
X = self.ambient_space().subscheme(self.Jacobian())
return X.rational_points(0, F)
X = self.singular_subscheme()
return X.rational_points(F=F)

def is_singular(self, P=None):
r"""
Return whether ``P`` is a singular point of this curve, or if no point is passed,
whether this curve is singular or not.
This just uses the is_smooth function for algebraic subschemes.
INPUT:
- ``P`` -- (default: None) a point on this curve.
OUTPUT:
- Boolean. If a point ``P`` is provided, and if ``P`` lies on this curve, returns True
if ``P`` is a singular point of this curve, and False otherwise. If no point is provided,
returns True or False depending on whether this curve is or is not singular, respectively.
EXAMPLES::
sage: P.<x,y,z,w> = ProjectiveSpace(QQ, 3)
sage: C = P.curve([y^2 - x^2 - z^2, z - w])
sage: C.is_singular()
False
::
sage: A.<x,y,z> = AffineSpace(GF(11), 3)
sage: C = A.curve([y^3 - z^5, x^5 - y + 1])
sage: Q = A([7,0,0])
sage: C.is_singular(Q)
True
"""
return not self.is_smooth(P)

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