def toPolynom(self,n):
"""
Return a couple compoosed by a polynomial ring and the polynomial
equal to the truncated series of degree n-1
EXAMPLES::
sage:
sage: L = u+v^2+3*w*z+w^2+u*v*z*w+u^2*v+w*z^4+v^6
sage: (PolRing,PolL) = L.toPolynom(3)
sage: PolL
v^2 + w^2 + 3*w*z + u
sage: PolRing
Multivariate Polynomial Ring in u, v, w, z over Rational Field
"""
if n>0:
from sage.rings.polynomial.all import PolynomialRing
BR=self.parent().base_ring()
R = PolynomialRing(BR,self.parent()._names)
v = R.gens()
pol = R(0)
for i in range(n):
l = self.coefficient(i)
for (e,t) in l:
pol += e*reduce((lambda a,b:a*b),map((lambda a,b:a**b),v,t))
return R,pol
else:
raise ValueError, "n must be a nonnegative integer"
def compute_bd(f, b, df, r, alpha):
"""Determine the next integral basis element form those already computed."""
# obtain the ring of Puiseux series in which the truncated series
# live. these should already be such that the base ring is SR, the symbolic
# ring. (below we will have to introduce symbolic indeterminants)
R = f.parent()
F = R.fraction_field()
x,y = R.gens()
# construct a list of indeterminants and a guess for the next integral
# basis element. to make computations uniform in the univariate and
# multivariate cases an additional generator of the underlying polynomial
# ring is introduced.
d = len(b)
Q = PolynomialRing(QQbar, ['a%d'%n for n in range(d)] + ['dummy'])
a = tuple(Q.gens())
b = tuple(b)
P = PuiseuxSeriesRing(Q, str(x))
xx = P.gen()
bd = F(y*b[-1])
# XXX HACK
for l in range(len(r)):
for k in range(len(r[l])):
r[l][k] = r[l][k].change_ring(Q)
# sufficiently singularize the current integral basis element guess at each
# of the singular points of df
for l in range(len(df)):
k = df[l] # factor
alphak = alpha[l] # point at which the truncated series are centered
rk = r[l] # truncated puiseux series
# singularize the current guess at the current point using each
# truncated Puiseux seriesx
sufficiently_singular = False
while not sufficiently_singular:
# from each puiseux series, rki, centered at alphak construct a
# system of equations from the negative exponent terms appearing in
# the expression A(x,rki))
equations = []
for rki in rk:
# A = sum(a[j] * b[j](xx,rki) for j in range(d))
A = evaluate_A(a,b,xx,rki,d)
A += bd(xx, rki)
# implicit division by x-alphak, hence truncation to x^1
terms = A.truncate(1).coefficients()
equations.extend(terms)
# attempt to solve this linear system of equations. if a (unique)
# solution exists then the integral basis element is not singular
# enough at alphak
sols = solve_coefficient_system(Q, equations, a)
if not sols is None:
bdm1 = sum(F(sols[i][0])*b[i] for i in range(d))
bd = F(bdm1 + bd)/ F(k)
else:
sufficiently_singular = True
return bd
def compute_bd(f, b, df, r, alpha):
"""Determine the next integral basis element form those already computed."""
# obtain the ring of Puiseux series in which the truncated series
# live. these should already be such that the base ring is SR, the symbolic
# ring. (below we will have to introduce symbolic indeterminants)
R = f.parent()
F = R.fraction_field()
x, y = R.gens()
# construct a list of indeterminants and a guess for the next integral
# basis element. to make computations uniform in the univariate and
# multivariate cases an additional generator of the underlying polynomial
# ring is introduced.
d = len(b)
Q = PolynomialRing(QQbar, ['a%d' % n for n in range(d)] + ['dummy'])
a = tuple(Q.gens())
b = tuple(b)
P = PuiseuxSeriesRing(Q, str(x))
xx = P.gen()
bd = F(y * b[-1])
# XXX HACK
for l in range(len(r)):
for k in range(len(r[l])):
r[l][k] = r[l][k].change_ring(Q)
# sufficiently singularize the current integral basis element guess at each
# of the singular points of df
for l in range(len(df)):
k = df[l] # factor
alphak = alpha[l] # point at which the truncated series are centered
rk = r[l] # truncated puiseux series
# singularize the current guess at the current point using each
# truncated Puiseux seriesx
sufficiently_singular = False
while not sufficiently_singular:
# from each puiseux series, rki, centered at alphak construct a
# system of equations from the negative exponent terms appearing in
# the expression A(x,rki))
equations = []
for rki in rk:
# A = sum(a[j] * b[j](xx,rki) for j in range(d))
A = evaluate_A(a, b, xx, rki, d)
A += bd(xx, rki)
# implicit division by x-alphak, hence truncation to x^1
terms = A.truncate(1).coefficients()
equations.extend(terms)
# attempt to solve this linear system of equations. if a (unique)
# solution exists then the integral basis element is not singular
# enough at alphak
sols = solve_coefficient_system(Q, equations, a)
if not sols is None:
bdm1 = sum(F(sols[i][0]) * b[i] for i in range(d))
bd = F(bdm1 + bd) / F(k)
else:
sufficiently_singular = True
return bd
def specialize(self):
"""specializes to weight k -- i.e. projects to Sym^k"""
k=self.weight
if k==0:
# R.<X,Y>=PolynomialRing(QQ,2)
R = PolynomialRing(QQ,('X','Y'))
X,Y = R.gens()
P=0
for j in range(0,k+1):
P=P+binomial(k,j)*((-1)**j)*self.moment(j)*(X**j)*(Y**(k-j))
return symk(k,P)
def map(self,psi):
"""psi is a map from the base_ring to Qp and this function applies psi to all polynomial coefficients and then lifts them to QQ"""
#assert psi.domain()==self.base_ring
k=self.weight
# S.<X,Y>=PolynomialRing(QQ)
S = PolynomialRing(QQ,('X','Y'))
X,Y = S.gens()
ans=0*X
for j in range(k+1):
ans=ans+psi(self.coef(j)).lift()*(X**j)*(Y**(k-j))
temp=copy(self)
temp.poly=ans
temp.base_ring=psi.codomain()
return temp
def __init__(self,k,poly=None,base_ring=QQ):
"""A symk object is stored as a homogeneous polynomial of degree k
Inputs:
k -- weight
poly -- homogeneous of weight k in X and Y
base_ring -- ring containing the coefficients of poly"""
#assert (poly == None) or (poly == 0) or (k==0) or (poly.is_homogeneous()), "must enter a homogeneous polynomial"
#if (poly != None) and (poly !=0) and (k<>0):
# assert poly.total_degree() == k, "the weight is incorrect"
if poly != None:
self.poly=poly
else:
# R.<X,Y>=PolynomialRing(base_ring,2)
# self.poly=0*X
R = PolynomialRing(base_ring,('X','Y'))
self.poly=0*R.gens()[0]
self.weight=k
self.base_ring=base_ring
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: 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
