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 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: FF = FiniteField(5)
sage: P2 = ProjectiveSpace(2, FF, names = ['x','y','z'])
sage: x, y, z = P2.coordinate_ring().gens()
sage: C = Curve(y^2*z^7-x^9-x*z^8)
sage: pt = C([2,3,1])
sage: C.local_coordinates(pt,9) # todo: not implemented !!!!
[2 + t, 3 + 3*t^2 + t^3 + 3*t^4 + 3*t^6 + 3*t^7 + t^8 + 2*t^9 + 3*t^11 + 3*t^12]
"""
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))
cmd = 'matrix c = coeffs ('+str(ft)+',t)'
S.eval(cmd)
N = int(S.eval('size(c)'))
b = ["c["+str(i)+",1]," 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)
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 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 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: FF = FiniteField(5)
sage: P2 = ProjectiveSpace(2, FF, names = ['x','y','z'])
sage: x, y, z = P2.coordinate_ring().gens()
sage: C = Curve(y^2*z^7-x^9-x*z^8)
sage: pt = C([2,3,1])
sage: C.local_coordinates(pt,9) # todo: not implemented !!!!
[2 + t, 3 + 3*t^2 + t^3 + 3*t^4 + 3*t^6 + 3*t^7 + t^8 + 2*t^9 + 3*t^11 + 3*t^12]
"""
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))
cmd = 'matrix c = coeffs (' + str(ft) + ',t)'
S.eval(cmd)
N = int(S.eval('size(c)'))
b = ["c[" + str(i) + ",1]," 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)
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