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affine_curve.py
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affine_curve.py
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"""
Affine plane curves over a general ring
AUTHORS:
- William Stein (2005-11-13)
- David Joyner (2005-11-13)
- David Kohel (2006-01)
"""
#*****************************************************************************
# 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.categories.homset import Hom
from sage.interfaces.all import singular
from sage.misc.all import add
from sage.rings.all import degree_lowest_rational_function
from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
from sage.schemes.affine.affine_space import is_AffineSpace
from sage.schemes.generic.algebraic_scheme import AlgebraicScheme_subscheme_affine
from sage.schemes.projective.projective_space import ProjectiveSpace
from curve import Curve_generic
class AffineSpaceCurve_generic(Curve_generic, AlgebraicScheme_subscheme_affine):
def _repr_type(self):
return "Affine Space"
def __init__(self, A, X):
if not is_AffineSpace(A):
raise TypeError("A (=%s) must be an affine space"%A)
Curve_generic.__init__(self, A, X)
d = self.dimension()
if d != 1:
raise ValueError("defining equations (=%s) define a scheme of dimension %s != 1"%(X,d))
def projective_closure(self, i=0):
r"""
Return the projective closure of this affine curve.
INPUT:
- ``i`` - the index of the affine coordinate chart of the projective space that the affine ambient space
of this curve embeds into.
OUTPUT:
- a curve in projective space.
EXAMPLES::
sage: A.<x,y,z> = AffineSpace(CC,3)
sage: C = Curve([x-y,z-2])
sage: C.projective_closure()
Projective Space Curve over Complex Field with 53 bits of precision defined by
x1 - x2, (-2.00000000000000)*x0 + x3
::
sage: A.<x,y,z> = AffineSpace(QQ,3)
sage: C = Curve([y-x^2,z-x^3])
sage: C.projective_closure()
Projective Space Curve over Rational Field defined by
x1^2 - x0*x2, x1*x2 - x0*x3, x2^2 - x1*x3
"""
I = self.defining_ideal()
# compute a Groebner basis of this ideal with respect to a graded monomial order
R = self.ambient_space().coordinate_ring().change_ring(order='degrevlex')
P = self.ambient_space().projective_embedding(i).codomain()
RH = P.coordinate_ring()
G = R.ideal([R(f) for f in I.gens()]).groebner_basis()
H = Hom(R,RH)
l = list(RH.gens())
x = l.pop(i)
phi = H(l)
from constructor import Curve
return Curve([phi(f).homogenize(x) for f in G])
class AffineCurve_generic(AffineSpaceCurve_generic):
def __init__(self, A, f):
if not (is_AffineSpace(A) and A.dimension != 2):
raise TypeError("Argument A (= %s) must be an affine plane."%A)
Curve_generic.__init__(self, A, [f])
def _repr_type(self):
return "Affine"
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]
"""
f = self.defining_polynomial()
R = f.parent()
F = self.base_ring()
p = F.characteristic()
x0 = F(pt[0])
y0 = F(pt[1])
astr = ["a"+str(i) for i in range(1,2*n)]
x,y = R.gens()
R0 = PolynomialRing(F,2*n+2,names = [str(x),str(y),"t"]+astr)
vars0 = R0.gens()
t = vars0[2]
yt = y0*t**0+add([vars0[i]*t**(i-2) for i in range(3,2*n+2)])
xt = x0+t
ft = f(xt,yt)
S = singular
S.eval('ring s = '+str(p)+','+str(R0.gens())+',lp;')
S.eval('poly f = '+str(ft) + ';')
c = S('coeffs(%s, t)'%ft)
N = int(c.size())
b = ["%s[%s,1],"%(c.name(), i) for i in range(2,N//2-4)]
b = ''.join(b)
b = b[:len(b)-1] # to cut off the trailing comma
cmd = 'ideal I = '+b
S.eval(cmd)
S.eval('short=0') # print using *'s and ^'s.
c = S.eval('slimgb(I)')
d = c.split("=")
d = d[1:]
d[len(d)-1] += "\n"
e = [x[:x.index("\n")] for x in d]
vals = []
for x in e:
for y in vars0:
if str(y) in x:
if len(x.replace(str(y),"")) != 0:
i = x.find("-")
if i>0:
vals.append([eval(x[1:i]),x[:i],F(eval(x[i+1:]))])
i = x.find("+")
if i>0:
vals.append([eval(x[1:i]),x[:i],-F(eval(x[i+1:]))])
else:
vals.append([eval(str(y)[1:]),str(y),F(0)])
vals.sort()
k = len(vals)
v = [x0+t,y0+add([vals[i][2]*t**(i+1) for i in range(k)])]
return v
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
"""
I = self.defining_ideal()
return I.plot(*args, **kwds)
class AffineCurve_finite_field(AffineCurve_generic):
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.
EXAMPLE::
sage: A.<x,y> = AffineSpace(2,GF(9,'a'))
sage: C = Curve(x^2 + y^2 - 1)
sage: C
Affine 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)]
"""
f = self.defining_polynomial()
R = f.parent()
K = R.base_ring()
points = []
for x in K:
for y in K:
if f(x,y) == 0:
points.append(self((x,y)))
points.sort()
return points
class AffineCurve_prime_finite_field(AffineCurve_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)
EXAMPLE::
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]
"""
f = self.defining_polynomial()
R = f.parent()
F = self.base_ring()
p = F.characteristic()
Dstr = str(tuple(D))
G = singular(','.join([str(x) for x in D]), type='intvec')
singular.LIB('brnoeth.lib')
S = singular.ring(p, R.gens(), 'lp')
fsing = singular(str(f))
X = fsing.Adj_div()
P = singular.NSplaces(1, X)
T = P[1][2]
T.set_ring()
LG = G.BrillNoether(P)
dim = len(LG)
basis = [(LG[i][1], LG[i][2]) for i in range(1,dim+1)]
x, y, z = PolynomialRing(F, 3, names = ["x","y","z"]).gens()
V = []
for g in basis:
T.set_ring() # necessary...
V.append(eval(g[0].sage_polystring())/eval(g[1].sage_polystring()))
return V
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)