def genus(self):
"""
Return the genus of this function field
For now, the genus is computed using singular
EXAMPLES::
sage: K.<x> = FunctionField(QQ); R.<y> = K[]
sage: L.<y> = K.extension(y^3 - (x^3 + 2*x*y + 1/x))
sage: L.genus()
3
"""
# unfortunately singular can not compute the genus with the polynomial_ring()._singular_
# object because genus method only accepts a ring of transdental degree 2 over a prime field
# not a ring of transdental degree 1 over a rational function field of one variable
if is_RationalFunctionField(self._base_field) and self._base_field.constant_field().is_prime_field():
#Making the auxiliary ring which only has polynomials with integral coefficients.
tmpAuxRing = PolynomialRing(self._base_field.constant_field(), str(self._base_field.gen())+','+str(self._ring.gen()))
intMinPoly, d = self._make_monic_integral(self._polynomial)
curveIdeal = tmpAuxRing.ideal(intMinPoly)
singular.lib('normal.lib') #loading genus method in singular
return int(curveIdeal._singular_().genus())
else:
raise NotImplementedError("Computation of genus over this rational function field not implemented yet")
def genus(self):
"""
Return the genus of this function field
For now, the genus is computed using singular
EXAMPLES::
sage: K.<x> = FunctionField(QQ); R.<y> = K[]
sage: L.<y> = K.extension(y^3 - (x^3 + 2*x*y + 1/x))
sage: L.genus()
3
"""
# unfortunately singular can not compute the genus with the polynomial_ring()._singular_
# object because genus method only accepts a ring of transdental degree 2 over a prime field
# not a ring of transdental degree 1 over a rational function field of one variable
if is_RationalFunctionField(
self._base_field) and self._base_field.constant_field(
).is_prime_field():
#Making the auxiliary ring which only has polynomials with integral coefficients.
tmpAuxRing = PolynomialRing(
self._base_field.constant_field(),
str(self._base_field.gen()) + ',' + str(self._ring.gen()))
intMinPoly, d = self._make_monic_integral(self._polynomial)
curveIdeal = tmpAuxRing.ideal(intMinPoly)
singular.lib('normal.lib') #loading genus method in singular
return int(curveIdeal._singular_().genus())
else:
raise NotImplementedError(
"Computation of genus over this rational function field not implemented yet"
)
def is_complete_intersection(self):
r"""
Return whether this projective curve is or is not a complete intersection.
OUTPUT: Boolean.
EXAMPLES::
sage: P.<x,y,z,w> = ProjectiveSpace(QQ, 3)
sage: C = Curve([x*y - z*w, x^2 - y*w, y^2*w - x*z*w], P)
sage: C.is_complete_intersection()
False
::
sage: P.<x,y,z,w> = ProjectiveSpace(QQ, 3)
sage: C = Curve([y*w - x^2, z*w^2 - x^3], P)
sage: C.is_complete_intersection()
True
sage: P.<x,y,z,w> = ProjectiveSpace(QQ, 3)
sage: C = Curve([z^2 - y*w, y*z - x*w, y^2 - x*z], P)
sage: C.is_complete_intersection()
False
"""
singular.lib("sing.lib")
I = singular.simplify(self.defining_ideal(), 10)
L = singular.is_ci(I).sage()
return len(self.ambient_space().gens()) - len(I.sage().gens()) == L[-1]
def is_complete_intersection(self):
r"""
Return whether this projective curve is or is not a complete intersection.
OUTPUT: Boolean.
EXAMPLES::
sage: P.<x,y,z,w> = ProjectiveSpace(QQ, 3)
sage: C = Curve([x*y - z*w, x^2 - y*w, y^2*w - x*z*w], P)
sage: C.is_complete_intersection()
False
::
sage: P.<x,y,z,w> = ProjectiveSpace(QQ, 3)
sage: C = Curve([y*w - x^2, z*w^2 - x^3], P)
sage: C.is_complete_intersection()
True
sage: P.<x,y,z,w> = ProjectiveSpace(QQ, 3)
sage: C = Curve([z^2 - y*w, y*z - x*w, y^2 - x*z], P)
sage: C.is_complete_intersection()
False
"""
singular.lib("sing.lib")
I = singular.simplify(self.defining_ideal(), 10)
L = singular.is_ci(I).sage()
return len(self.ambient_space().gens()) - len(I.sage().gens()) == L[-1]
def riemann_roch_basis(self, D):
r"""
Return a basis for the Riemann-Roch space corresponding to
`D`.
This uses Singular's Brill-Noether implementation.
INPUT:
- ``D`` - a divisor
OUTPUT:
A list of function field elements that form a basis of the Riemann-Roch space
EXAMPLE::
sage: R.<x,y,z> = GF(2)[]
sage: f = x^3*y + y^3*z + x*z^3
sage: C = Curve(f); pts = C.rational_points()
sage: D = C.divisor([ (4, pts[0]), (4, pts[2]) ])
sage: C.riemann_roch_basis(D)
[x/y, 1, z/y, z^2/y^2, z/x, z^2/(x*y)]
::
sage: R.<x,y,z> = GF(5)[]
sage: f = x^7 + y^7 + z^7
sage: C = Curve(f); pts = C.rational_points()
sage: D = C.divisor([ (3, pts[0]), (-1,pts[1]), (10, pts[5]) ])
sage: C.riemann_roch_basis(D)
[(-2*x + y)/(x + y), (-x + z)/(x + y)]
.. NOTE::
Currently this only works over prime field and divisors supported on rational points.
"""
f = self.defining_polynomial()._singular_()
singular = f.parent()
singular.lib('brnoeth')
try:
X1 = f.Adj_div()
except (TypeError, RuntimeError), s:
raise RuntimeError, str(
s
) + "\n\n ** Unable to use the Brill-Noether Singular package to compute all points (see above)."
def riemann_roch_basis(self, D):
r"""
Return a basis for the Riemann-Roch space corresponding to
`D`.
This uses Singular's Brill-Noether implementation.
INPUT:
- ``D`` - a divisor
OUTPUT:
A list of function field elements that form a basis of the Riemann-Roch space
EXAMPLE::
sage: R.<x,y,z> = GF(2)[]
sage: f = x^3*y + y^3*z + x*z^3
sage: C = Curve(f); pts = C.rational_points()
sage: D = C.divisor([ (4, pts[0]), (4, pts[2]) ])
sage: C.riemann_roch_basis(D)
[x/y, 1, z/y, z^2/y^2, z/x, z^2/(x*y)]
::
sage: R.<x,y,z> = GF(5)[]
sage: f = x^7 + y^7 + z^7
sage: C = Curve(f); pts = C.rational_points()
sage: D = C.divisor([ (3, pts[0]), (-1,pts[1]), (10, pts[5]) ])
sage: C.riemann_roch_basis(D)
[(-2*x + y)/(x + y), (-x + z)/(x + y)]
.. NOTE::
Currently this only works over prime field and divisors supported on rational points.
"""
f = self.defining_polynomial()._singular_()
singular = f.parent()
singular.lib('brnoeth')
try:
X1 = f.Adj_div()
except (TypeError, RuntimeError), s:
raise RuntimeError, str(s) + "\n\n ** Unable to use the Brill-Noether Singular package to compute all points (see above)."
def _points_via_singular(self, sort=True):
r"""
Return all rational points on this curve, computed using Singular's
Brill-Noether implementation.
INPUT:
- ``sort`` - bool (default: True), if True return the
point list sorted. If False, returns the points in the order
computed by Singular.
EXAMPLE::
sage: x, y, z = PolynomialRing(GF(5), 3, 'xyz').gens()
sage: f = y^2*z^7 - x^9 - x*z^8
sage: C = Curve(f); C
Projective Curve over Finite Field of size 5 defined by
-x^9 + y^2*z^7 - x*z^8
sage: C._points_via_singular()
[(0 : 0 : 1), (0 : 1 : 0), (2 : 2 : 1), (2 : 3 : 1),
(3 : 1 : 1), (3 : 4 : 1)]
sage: C._points_via_singular(sort=False) #random
[(0 : 1 : 0), (3 : 1 : 1), (3 : 4 : 1), (2 : 2 : 1),
(0 : 0 : 1), (2 : 3 : 1)]
.. note::
The Brill-Noether package does not always work (i.e., the
'bn' algorithm. When it fails a RuntimeError exception is
raised.
"""
f = self.defining_polynomial()._singular_()
singular = f.parent()
singular.lib('brnoeth')
try:
X1 = f.Adj_div()
except (TypeError, RuntimeError), s:
raise RuntimeError, str(s) + "\n\n ** Unable to use the\
def _points_via_singular(self, sort=True):
r"""
Return all rational points on this curve, computed using Singular's
Brill-Noether implementation.
INPUT:
- ``sort`` - bool (default: True), if True return the
point list sorted. If False, returns the points in the order
computed by Singular.
EXAMPLE::
sage: x, y, z = PolynomialRing(GF(5), 3, 'xyz').gens()
sage: f = y^2*z^7 - x^9 - x*z^8
sage: C = Curve(f); C
Projective Curve over Finite Field of size 5 defined by
-x^9 + y^2*z^7 - x*z^8
sage: C._points_via_singular()
[(0 : 0 : 1), (0 : 1 : 0), (2 : 2 : 1), (2 : 3 : 1),
(3 : 1 : 1), (3 : 4 : 1)]
sage: C._points_via_singular(sort=False) #random
[(0 : 1 : 0), (3 : 1 : 1), (3 : 4 : 1), (2 : 2 : 1),
(0 : 0 : 1), (2 : 3 : 1)]
.. note::
The Brill-Noether package does not always work (i.e., the
'bn' algorithm. When it fails a RuntimeError exception is
raised.
"""
f = self.defining_polynomial()._singular_()
singular = f.parent()
singular.lib('brnoeth')
try:
X1 = f.Adj_div()
except (TypeError, RuntimeError), s:
raise RuntimeError, str(s) + "\n\n ** Unable to use the\
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.
EXAMPLE::
sage: x, y = (GF(5)['x,y']).gens()
sage: f = y^2 - x^9 - x
sage: C = Curve(f); C
Affine 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
"""
if algorithm == "enum":
return AffineCurve_finite_field.rational_points(self,
algorithm="enum")
elif algorithm == "bn":
f = self.defining_polynomial()._singular_()
singular = f.parent()
singular.lib('brnoeth')
try:
X1 = f.Adj_div()
except (TypeError, RuntimeError), s:
raise RuntimeError, str(
s
) + "\n\n ** Unable to use the Brill-Noether Singular package to compute all points (see above)."
X2 = singular.NSplaces(1, X1)
R = X2[5][1][1]
singular.set_ring(R)
# 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).
v = singular('POINTS').sage_flattened_str_list()
pnts = [
self(int(v[3 * i]), int(v[3 * i + 1]))
for i in range(len(v) / 3) if int(v[3 * i + 2]) != 0
]
# remove multiple points
pnts = list(set(pnts))
pnts.sort()
return pnts
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.
EXAMPLE::
sage: x, y = (GF(5)['x,y']).gens()
sage: f = y^2 - x^9 - x
sage: C = Curve(f); C
Affine 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
"""
if algorithm == "enum":
return AffineCurve_finite_field.rational_points(self, algorithm="enum")
elif algorithm == "bn":
f = self.defining_polynomial()._singular_()
singular = f.parent()
singular.lib('brnoeth')
try:
X1 = f.Adj_div()
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).")
X2 = singular.NSplaces(1, X1)
R = X2[5][1][1]
singular.set_ring(R)
# 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).
v = singular('POINTS').sage_flattened_str_list()
pnts = [self(int(v[3*i]), int(v[3*i+1])) for i in range(len(v)/3) if int(v[3*i+2])!=0]
# remove multiple points
pnts = sorted(set(pnts))
return pnts
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)
def riemann_roch_basis(self, D):
r"""
Return a basis for the Riemann-Roch space corresponding to
`D`.
This uses Singular's Brill-Noether implementation.
INPUT:
- ``D`` - a divisor
OUTPUT:
A list of function field elements that form a basis of the Riemann-Roch space
EXAMPLE::
sage: R.<x,y,z> = GF(2)[]
sage: f = x^3*y + y^3*z + x*z^3
sage: C = Curve(f); pts = C.rational_points()
sage: D = C.divisor([ (4, pts[0]), (4, pts[2]) ])
sage: C.riemann_roch_basis(D)
[x/y, 1, z/y, z^2/y^2, z/x, z^2/(x*y)]
::
sage: R.<x,y,z> = GF(5)[]
sage: f = x^7 + y^7 + z^7
sage: C = Curve(f); pts = C.rational_points()
sage: D = C.divisor([ (3, pts[0]), (-1,pts[1]), (10, pts[5]) ])
sage: C.riemann_roch_basis(D)
[(-2*x + y)/(x + y), (-x + z)/(x + y)]
.. NOTE::
Currently this only works over prime field and divisors supported on rational points.
"""
f = self.defining_polynomial()._singular_()
singular = f.parent()
singular.lib('brnoeth')
try:
X1 = f.Adj_div()
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).")
X2 = singular.NSplaces(1, X1)
# retrieve list of all computed closed points (possibly of degree >1)
v = X2[3].sage_flattened_str_list() # 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).
v = [ v[i].partition(',') for i in range(len(v)) ]
pnts = [ ( int(v[i][0]), int(v[i][2])-1 ) for i in range(len(v))]
# retrieve coordinates of rational points
R = X2[5][1][1]
singular.set_ring(R)
v = singular('POINTS').sage_flattened_str_list()
coords = [self(int(v[3*i]), int(v[3*i+1]), int(v[3*i+2])) for i in range(len(v)//3)]
# build correct representation of D for singular
Dsupport = D.support()
Dcoeffs = []
for x in pnts:
if x[0] == 1:
Dcoeffs.append(D.coefficient(coords[x[1]]))
else:
Dcoeffs.append(0)
Dstr = str(tuple(Dcoeffs))
G = singular(','.join([str(x) for x in Dcoeffs]), type='intvec')
# call singular's brill noether routine and return
T = X2[1][2]
T.set_ring()
LG = G.BrillNoether(X2)
LG = [X.split(',\n') for X in LG.sage_structured_str_list()]
x,y,z = self.ambient_space().coordinate_ring().gens()
vars = {'x':x, 'y':y, 'z':z}
V = [(sage_eval(a, vars)/sage_eval(b, vars)) for a, b in LG]
return V
def _points_via_singular(self, sort=True):
r"""
Return all rational points on this curve, computed using Singular's
Brill-Noether implementation.
INPUT:
- ``sort`` - bool (default: True), if True return the
point list sorted. If False, returns the points in the order
computed by Singular.
EXAMPLE::
sage: x, y, z = PolynomialRing(GF(5), 3, 'xyz').gens()
sage: f = y^2*z^7 - x^9 - x*z^8
sage: C = Curve(f); C
Projective Curve over Finite Field of size 5 defined by
-x^9 + y^2*z^7 - x*z^8
sage: C._points_via_singular()
[(0 : 0 : 1), (0 : 1 : 0), (2 : 2 : 1), (2 : 3 : 1),
(3 : 1 : 1), (3 : 4 : 1)]
sage: C._points_via_singular(sort=False) #random
[(0 : 1 : 0), (3 : 1 : 1), (3 : 4 : 1), (2 : 2 : 1),
(0 : 0 : 1), (2 : 3 : 1)]
.. note::
The Brill-Noether package does not always work (i.e., the
'bn' algorithm. When it fails a RuntimeError exception is
raised.
"""
f = self.defining_polynomial()._singular_()
singular = f.parent()
singular.lib('brnoeth')
try:
X1 = f.Adj_div()
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).")
X2 = singular.NSplaces(1, X1)
R = X2[5][1][1]
singular.set_ring(R)
# 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).
v = singular('POINTS').sage_flattened_str_list()
pnts = [self(int(v[3*i]), int(v[3*i+1]), int(v[3*i+2]))
for i in range(len(v)//3)]
# singular always dehomogenizes with respect to the last variable
# so if this variable divides the curve equation, we need to add
# points at infinity
F = self.defining_polynomial()
z = F.parent().gens()[-1]
if z.divides(F):
pnts += [self(1,a,0) for a in self.base_ring()]
pnts += [self(0,1,0)]
# remove multiple points
pnts = list(set(pnts))
if sort:
pnts.sort()
return pnts
def riemann_roch_basis(self, D):
r"""
Return a basis for the Riemann-Roch space corresponding to
`D`.
This uses Singular's Brill-Noether implementation.
INPUT:
- ``D`` - a divisor
OUTPUT:
A list of function field elements that form a basis of the Riemann-Roch space
EXAMPLE::
sage: R.<x,y,z> = GF(2)[]
sage: f = x^3*y + y^3*z + x*z^3
sage: C = Curve(f); pts = C.rational_points()
sage: D = C.divisor([ (4, pts[0]), (4, pts[2]) ])
sage: C.riemann_roch_basis(D)
[x/y, 1, z/y, z^2/y^2, z/x, z^2/(x*y)]
::
sage: R.<x,y,z> = GF(5)[]
sage: f = x^7 + y^7 + z^7
sage: C = Curve(f); pts = C.rational_points()
sage: D = C.divisor([ (3, pts[0]), (-1,pts[1]), (10, pts[5]) ])
sage: C.riemann_roch_basis(D)
[(-2*x + y)/(x + y), (-x + z)/(x + y)]
.. NOTE::
Currently this only works over prime field and divisors supported on rational points.
"""
f = self.defining_polynomial()._singular_()
singular = f.parent()
singular.lib('brnoeth')
try:
X1 = f.Adj_div()
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)."
)
X2 = singular.NSplaces(1, X1)
# retrieve list of all computed closed points (possibly of degree >1)
v = X2[3].sage_flattened_str_list(
) # 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).
v = [v[i].partition(',') for i in range(len(v))]
pnts = [(int(v[i][0]), int(v[i][2]) - 1) for i in range(len(v))]
# retrieve coordinates of rational points
R = X2[5][1][1]
singular.set_ring(R)
v = singular('POINTS').sage_flattened_str_list()
coords = [
self(int(v[3 * i]), int(v[3 * i + 1]), int(v[3 * i + 2]))
for i in range(len(v) // 3)
]
# build correct representation of D for singular
Dsupport = D.support()
Dcoeffs = []
for x in pnts:
if x[0] == 1:
Dcoeffs.append(D.coefficient(coords[x[1]]))
else:
Dcoeffs.append(0)
Dstr = str(tuple(Dcoeffs))
G = singular(','.join([str(x) for x in Dcoeffs]), type='intvec')
# call singular's brill noether routine and return
T = X2[1][2]
T.set_ring()
LG = G.BrillNoether(X2)
LG = [X.split(',\n') for X in LG.sage_structured_str_list()]
x, y, z = self.ambient_space().coordinate_ring().gens()
vars = {'x': x, 'y': y, 'z': z}
V = [(sage_eval(a, vars) / sage_eval(b, vars)) for a, b in LG]
return V
def _points_via_singular(self, sort=True):
r"""
Return all rational points on this curve, computed using Singular's
Brill-Noether implementation.
INPUT:
- ``sort`` - bool (default: True), if True return the
point list sorted. If False, returns the points in the order
computed by Singular.
EXAMPLE::
sage: x, y, z = PolynomialRing(GF(5), 3, 'xyz').gens()
sage: f = y^2*z^7 - x^9 - x*z^8
sage: C = Curve(f); C
Projective Curve over Finite Field of size 5 defined by
-x^9 + y^2*z^7 - x*z^8
sage: C._points_via_singular()
[(0 : 0 : 1), (0 : 1 : 0), (2 : 2 : 1), (2 : 3 : 1),
(3 : 1 : 1), (3 : 4 : 1)]
sage: C._points_via_singular(sort=False) #random
[(0 : 1 : 0), (3 : 1 : 1), (3 : 4 : 1), (2 : 2 : 1),
(0 : 0 : 1), (2 : 3 : 1)]
.. note::
The Brill-Noether package does not always work (i.e., the
'bn' algorithm. When it fails a RuntimeError exception is
raised.
"""
f = self.defining_polynomial()._singular_()
singular = f.parent()
singular.lib('brnoeth')
try:
X1 = f.Adj_div()
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).")
X2 = singular.NSplaces(1, X1)
R = X2[5][1][1]
singular.set_ring(R)
# 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).
v = singular('POINTS').sage_flattened_str_list()
pnts = [
self(int(v[3 * i]), int(v[3 * i + 1]), int(v[3 * i + 2]))
for i in range(len(v) // 3)
]
# singular always dehomogenizes with respect to the last variable
# so if this variable divides the curve equation, we need to add
# points at infinity
F = self.defining_polynomial()
z = F.parent().gens()[-1]
if z.divides(F):
pnts += [self(1, a, 0) for a in self.base_ring()]
pnts += [self(0, 1, 0)]
# remove multiple points
pnts = list(set(pnts))
if sort:
pnts.sort()
return pnts
def intersection_multiplicity(self, X, P):
r"""
Return the intersection multiplicity of this subscheme and the subscheme ``X`` at the point ``P``.
The intersection of this subscheme with ``X`` must be proper, that is `\mathrm{codim}(self\cap
X) = \mathrm{codim}(self) + \mathrm{codim}(X)`, and must also be finite. We use Serre's Tor
formula to compute the intersection multiplicity. If `I`, `J` are the defining ideals of ``self``, ``X``,
respectively, then this is `\sum_{i=0}^{\infty}(-1)^i\mathrm{length}(\mathrm{Tor}_{\mathcal{O}_{A,p}}^{i}
(\mathcal{O}_{A,p}/I,\mathcal{O}_{A,p}/J))` where `A` is the affine ambient space of these subschemes.
INPUT:
- ``X`` -- subscheme in the same ambient space as this subscheme.
- ``P`` -- a point in the intersection of this subscheme with ``X``.
OUTPUT: An integer.
EXAMPLES::
sage: A.<x,y> = AffineSpace(QQ, 2)
sage: C = Curve([y^2 - x^3 - x^2], A)
sage: D = Curve([y^2 + x^3], A)
sage: Q = A([0,0])
sage: C.intersection_multiplicity(D, Q)
4
::
sage: R.<a> = QQ[]
sage: K.<b> = NumberField(a^6 - 3*a^5 + 5*a^4 - 5*a^3 + 5*a^2 - 3*a + 1)
sage: A.<x,y,z,w> = AffineSpace(K, 4)
sage: X = A.subscheme([x*y, y*z + 7, w^3 - x^3])
sage: Y = A.subscheme([x - z^3 + z + 1])
sage: Q = A([0, -7*b^5 + 21*b^4 - 28*b^3 + 21*b^2 - 21*b + 14, -b^5 + 2*b^4 - 3*b^3 \
+ 2*b^2 - 2*b, 0])
sage: X.intersection_multiplicity(Y, Q)
3
::
sage: A.<x,y,z> = AffineSpace(QQ, 3)
sage: X = A.subscheme([z^2 - 1])
sage: Y = A.subscheme([z - 1, y - x^2])
sage: Q = A([1,1,1])
sage: X.intersection_multiplicity(Y, Q)
Traceback (most recent call last):
...
TypeError: the intersection of this subscheme and (=Closed subscheme of Affine Space of dimension 3
over Rational Field defined by: z - 1, -x^2 + y) must be proper and finite
::
sage: A.<x,y,z,w,t> = AffineSpace(QQ, 5)
sage: X = A.subscheme([x*y, t^2*w, w^3*z])
sage: Y = A.subscheme([y*w + z])
sage: Q = A([0,0,0,0,0])
sage: X.intersection_multiplicity(Y, Q)
Traceback (most recent call last):
...
TypeError: the intersection of this subscheme and (=Closed subscheme of Affine Space of dimension 5
over Rational Field defined by: y*w + z) must be proper and finite
"""
AA = self.ambient_space()
if AA != X.ambient_space():
raise TypeError(
"this subscheme and (=%s) must be defined in the same ambient space"
% X)
W = self.intersection(X)
try:
W._check_satisfies_equations(P)
except TypeError:
raise TypeError(
"(=%s) must be a point in the intersection of this subscheme and (=%s)"
% (P, X))
if AA.dimension() != self.dimension() + X.dimension() or W.dimension(
) != 0:
raise TypeError(
"the intersection of this subscheme and (=%s) must be proper and finite"
% X)
I = self.defining_ideal()
J = X.defining_ideal()
# move P to the origin and localize
chng_coords = [
AA.gens()[i] + P[i] for i in range(AA.dimension_relative())
]
R = AA.coordinate_ring().change_ring(order="negdegrevlex")
Iloc = R.ideal([f(chng_coords) for f in I.gens()])
Jloc = R.ideal([f(chng_coords) for f in J.gens()])
# compute the intersection multiplicity with Serre's Tor formula using Singular
singular.lib("homolog.lib")
i = 0
s = 0
t = sum(singular.Tor(i, Iloc, Jloc).std().hilb(2).sage())
while t != 0:
s = s + ((-1)**i) * t
i = i + 1
t = sum(singular.Tor(i, Iloc, Jloc).std().hilb(2).sage())
return s
def intersection_multiplicity(self, X, P):
r"""
Return the intersection multiplicity of this subscheme and the subscheme ``X`` at the point ``P``.
The intersection of this subscheme with ``X`` must be proper, that is `\mathrm{codim}(self\cap
X) = \mathrm{codim}(self) + \mathrm{codim}(X)`, and must also be finite. We use Serre's Tor
formula to compute the intersection multiplicity. If `I`, `J` are the defining ideals of ``self``, ``X``,
respectively, then this is `\sum_{i=0}^{\infty}(-1)^i\mathrm{length}(\mathrm{Tor}_{\mathcal{O}_{A,p}}^{i}
(\mathcal{O}_{A,p}/I,\mathcal{O}_{A,p}/J))` where `A` is the affine ambient space of these subschemes.
INPUT:
- ``X`` -- subscheme in the same ambient space as this subscheme.
- ``P`` -- a point in the intersection of this subscheme with ``X``.
OUTPUT: An integer.
EXAMPLES::
sage: A.<x,y> = AffineSpace(QQ, 2)
sage: C = Curve([y^2 - x^3 - x^2], A)
sage: D = Curve([y^2 + x^3], A)
sage: Q = A([0,0])
sage: C.intersection_multiplicity(D, Q)
4
::
sage: R.<a> = QQ[]
sage: K.<b> = NumberField(a^6 - 3*a^5 + 5*a^4 - 5*a^3 + 5*a^2 - 3*a + 1)
sage: A.<x,y,z,w> = AffineSpace(K, 4)
sage: X = A.subscheme([x*y, y*z + 7, w^3 - x^3])
sage: Y = A.subscheme([x - z^3 + z + 1])
sage: Q = A([0, -7*b^5 + 21*b^4 - 28*b^3 + 21*b^2 - 21*b + 14, -b^5 + 2*b^4 - 3*b^3 \
+ 2*b^2 - 2*b, 0])
sage: X.intersection_multiplicity(Y, Q)
3
::
sage: A.<x,y,z> = AffineSpace(QQ, 3)
sage: X = A.subscheme([z^2 - 1])
sage: Y = A.subscheme([z - 1, y - x^2])
sage: Q = A([1,1,1])
sage: X.intersection_multiplicity(Y, Q)
Traceback (most recent call last):
...
TypeError: the intersection of this subscheme and (=Closed subscheme of Affine Space of dimension 3
over Rational Field defined by: z - 1, -x^2 + y) must be proper and finite
::
sage: A.<x,y,z,w,t> = AffineSpace(QQ, 5)
sage: X = A.subscheme([x*y, t^2*w, w^3*z])
sage: Y = A.subscheme([y*w + z])
sage: Q = A([0,0,0,0,0])
sage: X.intersection_multiplicity(Y, Q)
Traceback (most recent call last):
...
TypeError: the intersection of this subscheme and (=Closed subscheme of Affine Space of dimension 5
over Rational Field defined by: y*w + z) must be proper and finite
"""
AA = self.ambient_space()
if AA != X.ambient_space():
raise TypeError("this subscheme and (=%s) must be defined in the same ambient space"%X)
W = self.intersection(X)
try:
W._check_satisfies_equations(P)
except TypeError:
raise TypeError("(=%s) must be a point in the intersection of this subscheme and (=%s)"%(P,X))
if AA.dimension() != self.dimension() + X.dimension() or W.dimension() != 0:
raise TypeError("the intersection of this subscheme and (=%s) must be proper and finite"%X)
I = self.defining_ideal()
J = X.defining_ideal()
# move P to the origin and localize
chng_coords = [AA.gens()[i] + P[i] for i in range(AA.dimension_relative())]
R = AA.coordinate_ring().change_ring(order="negdegrevlex")
Iloc = R.ideal([f(chng_coords) for f in I.gens()])
Jloc = R.ideal([f(chng_coords) for f in J.gens()])
# compute the intersection multiplicity with Serre's Tor formula using Singular
singular.lib("homolog.lib")
i = 0
s = 0
t = sum(singular.Tor(i, Iloc, Jloc).std().hilb(2).sage())
while t != 0:
s = s + ((-1)**i)*t
i = i + 1
t = sum(singular.Tor(i, Iloc, Jloc).std().hilb(2).sage())
return s
