def eqn_list_to_curve_plot(L, rat_pts):
xpoly_rng = PolynomialRing(QQ, 'x')
poly_tup = [xpoly_rng(tup) for tup in L]
f = poly_tup[0]
h = poly_tup[1]
g = f + h**2 / 4
if len(g.real_roots()) == 0 and g(0) < 0:
return text(r"$X(\mathbb{R})=\emptyset$", (1, 1), fontsize=50)
X0 = [real(z[0]) for z in g.base_extend(CC).roots()
] + [real(z[0]) for z in g.derivative().base_extend(CC).roots()]
a, b = inflate_interval(min(X0), max(X0), 1.5)
groots = [a] + g.real_roots() + [b]
if b - a < 1e-7:
a = -3
b = 3
groots = [a, b]
ngints = len(groots) - 1
plotzones = []
npts = 100
for j in range(ngints):
c = groots[j]
d = groots[j + 1]
if g((c + d) / 2) < 0:
continue
(c, d) = inflate_interval(c, d, 1.1)
s = (d - c) / npts
u = c
yvals = []
for i in range(npts + 1):
v = g(u)
if v > 0:
v = sqrt(v)
w = -h(u) / 2
yvals.append(w + v)
yvals.append(w - v)
u += s
(m, M) = inflate_interval(min(yvals), max(yvals), 1.2)
plotzones.append((c, d, m, M))
x = var('x')
y = var('y')
plot = sum(
implicit_plot(y**2 + y * h(x) - f(x), (x, R[0], R[1]), (y, R[2], R[3]),
aspect_ratio='automatic',
plot_points=500,
zorder=1) for R in plotzones)
xmin = min([R[0] for R in plotzones])
xmax = max([R[1] for R in plotzones])
ymin = min([R[2] for R in plotzones])
ymax = max([R[3] for R in plotzones])
for P in rat_pts:
(x, y, z) = P
z = ZZ(z)
if z: # Do not attempt to plot points at infinity
x = ZZ(x) / z
y = ZZ(y) / z**3
if x >= xmin and x <= xmax and y >= ymin and y <= ymax:
plot += point((x, y), color='red', size=40, zorder=2)
return plot
def eqn_list_to_curve_plot(L,rat_pts):
xpoly_rng = PolynomialRing(QQ,'x')
poly_tup = [xpoly_rng(tup) for tup in L]
f = poly_tup[0]
h = poly_tup[1]
g = f+h**2/4
if len(g.real_roots())==0 and g(0)<0:
return text("$X(\mathbb{R})=\emptyset$",(1,1),fontsize=50)
X0 = [real(z[0]) for z in g.base_extend(CC).roots()]+[real(z[0]) for z in g.derivative().base_extend(CC).roots()]
a,b = inflate_interval(min(X0),max(X0),1.5)
groots = [a]+g.real_roots()+[b]
if b-a<1e-7:
a=-3
b=3
groots=[a,b]
ngints = len(groots)-1
plotzones = []
npts = 100
for j in range(ngints):
c = groots[j]
d = groots[j+1]
if g((c+d)/2)<0:
continue
(c,d) = inflate_interval(c,d,1.1)
s = (d-c)/npts
u = c
yvals = []
for i in range(npts+1):
v = g(u)
if v>0:
v = sqrt(v)
w = -h(u)/2
yvals.append(w+v)
yvals.append(w-v)
u += s
(m,M) = inflate_interval(min(yvals),max(yvals),1.2)
plotzones.append((c,d,m,M))
x = var('x')
y = var('y')
plot=sum(implicit_plot(y**2 + y*h(x) - f(x), (x,R[0],R[1]),(y,R[2],R[3]), aspect_ratio='automatic', plot_points=500, zorder=1) for R in plotzones)
xmin=min([R[0] for R in plotzones])
xmax=max([R[1] for R in plotzones])
ymin=min([R[2] for R in plotzones])
ymax=max([R[3] for R in plotzones])
for P in rat_pts:
(x,y,z)=eval(P.replace(':',','))
z=ZZ(z)
if z: # Do not attempt to plot points at infinity
x=ZZ(x)/z
y=ZZ(y)/z**3
if x >= xmin and x <= xmax and y >= ymin and y <= ymax:
plot += point((x,y),color='red',size=40,zorder=2)
return plot
def eqn_list_to_curve_plot(L):
xpoly_rng = PolynomialRing(QQ, 'x')
poly_tup = [xpoly_rng(tup) for tup in L]
f = poly_tup[0]
h = poly_tup[1]
g = f + h**2 / 4
if len(g.real_roots()) == 0 and g(0) < 0:
return text("$X(\mathbb{R})=\emptyset$", (1, 1), fontsize=50)
X0 = [real(z[0]) for z in g.base_extend(CC).roots()
] + [real(z[0]) for z in g.derivative().base_extend(CC).roots()]
a, b = inflate_interval(min(X0), max(X0), 1.5)
groots = [a] + g.real_roots() + [b]
if b - a < 1e-7:
a = -3
b = 3
groots = [a, b]
ngints = len(groots) - 1
plotzones = []
npts = 100
for j in range(ngints):
c = groots[j]
d = groots[j + 1]
if g((c + d) / 2) < 0:
continue
(c, d) = inflate_interval(c, d, 1.1)
s = (d - c) / npts
u = c
yvals = []
for i in range(npts + 1):
v = g(u)
if v > 0:
v = sqrt(v)
w = -h(u) / 2
yvals.append(w + v)
yvals.append(w - v)
u += s
(m, M) = inflate_interval(min(yvals), max(yvals), 1.2)
plotzones.append((c, d, m, M))
x = var('x')
y = var('y')
return sum(
implicit_plot(y**2 + y * h(x) - f(x), (x, R[0], R[1]), (y, R[2], R[3]),
aspect_ratio='automatic',
plot_points=500) for R in plotzones)
def eqn_list_to_curve_plot(L):
xpoly_rng = PolynomialRing(QQ,'x')
poly_tup = [xpoly_rng(tup) for tup in L]
f = poly_tup[0]
h = poly_tup[1]
g = f+h**2/4
if len(g.real_roots())==0 and g(0)<0:
return text("$X(\mathbb{R})=\emptyset$",(1,1),fontsize=50)
X0 = [real(z[0]) for z in g.base_extend(CC).roots()]+[real(z[0]) for z in g.derivative().base_extend(CC).roots()]
a,b = inflate_interval(min(X0),max(X0),1.5)
groots = [a]+g.real_roots()+[b]
if b-a<1e-7:
a=-3
b=3
groots=[a,b]
ngints = len(groots)-1
plotzones = []
npts = 100
for j in range(ngints):
c = groots[j]
d = groots[j+1]
if g((c+d)/2)<0:
continue
(c,d) = inflate_interval(c,d,1.1)
s = (d-c)/npts
u = c
yvals = []
for i in range(npts+1):
v = g(u)
if v>0:
v = sqrt(v)
w = -h(u)/2
yvals.append(w+v)
yvals.append(w-v)
u += s
(m,M) = inflate_interval(min(yvals),max(yvals),1.2)
plotzones.append((c,d,m,M))
x = var('x')
y = var('y')
return sum(implicit_plot(y**2 + y*h(x) - f(x), (x,R[0],R[1]),
(y,R[2],R[3]), aspect_ratio='automatic', plot_points=500) for R in
plotzones)
def dimension(self,k,ignore=False, debug = 0):
if k < 2 and not ignore:
raise NotImplementedError("k has to >= 2")
s = self._signature
if not (2*k in ZZ):
raise ValueError("k has to be integral or half-integral")
if (2*k+s)%4 != 0 and not ignore:
raise NotImplementedError("2k has to be congruent to -signature mod 4")
if self._v2.has_key(0):
v2 = self._v2[0]
else:
v2 = 1
if self._g != None:
if not self._aniso_formula:
vals = self._g.values()
#else:
#print "using aniso_formula"
M = self._g
else:
vals = self._M.values()
M = self._M
prec = ceil(max(log(M.order(),2),52)+1)+17
#print prec
RR = RealField(prec)
CC = ComplexField(prec)
d = self._d
m = self._m
if debug > 0: print d,m
if self._alpha3 == None:
if self._aniso_formula:
self._alpha4 = 1
self._alpha3 = -sum([BB(a)*mm for a,mm in self._v2.iteritems() if a != 0])
#print self._alpha3
self._alpha3 += Integer(d) - Integer(1) - self._g.beta_formula()
#print self._alpha3, self._g.a5prime_formula()
self._alpha3 = self._alpha3/RR(2)
else:
self._alpha3 = sum([(1-a)*mm for a,mm in self._v2.iteritems() if a != 0])
#print self._alpha3
self._alpha3 += sum([(1-a)*mm for a,mm in vals.iteritems() if a != 0])
#print self._alpha3
self._alpha3 = self._alpha3 / Integer(2)
self._alpha4 = 1/Integer(2)*(vals[0]+v2) # the codimension of SkL in MkL
alpha3 = self._alpha3
alpha4 = self._alpha4
if debug > 0: print alpha3, alpha4
g1=M.char_invariant(1)
g1=CC(g1[0]*g1[1])
#print g1
g2=M.char_invariant(2)
g2=RR(real(g2[0]*g2[1]))
if debug > 0: print g2, g2.parent()
g3=M.char_invariant(-3)
g3=CC(g3[0]*g3[1])
if debug > 0: print RR(d) / RR(4), sqrt(RR(m)) / RR(4), CC(exp(2 * CC.pi() * CC(0,1) * (2 * k + s) / Integer(8)))
alpha1 = RR(d) / RR(4) - (sqrt(RR(m)) / RR(4) * CC(exp(2 * CC.pi() * CC(0,1) * (2 * k + s) / Integer(8))) * g2)
if debug > 0: print alpha1
alpha2 = RR(d) / RR(3) + sqrt(RR(m)) / (3 * sqrt(RR(3))) * real(exp(CC(2 * CC.pi() * CC(0,1) * (4 * k + 3 * s - 10) / 24)) * (g1+g3))
if debug > 0: print alpha1, alpha2, g1, g2, g3, d, k, s
dim = real(d + (d * k / Integer(12)) - alpha1 - alpha2 - alpha3)
if debug > 0:
print "dimension:", dim
if abs(dim-round(dim)) > 1e-6:
raise RuntimeError("Error ({0}) too large in dimension formula for {1} and k={2}".format(abs(dim-round(dim)), self._M if self._M is not None else self._g, k))
dimr = dim
dim = Integer(round(dim))
if k >=2 and dim < 0:
raise RuntimeError("Negative dimension (= {0}, {1})!".format(dim, dimr))
return dim
def dimension(self, k, ignore=False, debug=0):
if k < 2 and not ignore:
raise NotImplementedError("k has to >= 2")
s = self._signature
if not (2 * k in ZZ):
raise ValueError("k has to be integral or half-integral")
if (2 * k + s) % 4 != 0 and not ignore:
raise NotImplementedError(
"2k has to be congruent to -signature mod 4")
if self._v2.has_key(0):
v2 = self._v2[0]
else:
v2 = 1
if self._g != None:
if not self._aniso_formula:
vals = self._g.values()
#else:
#print "using aniso_formula"
M = self._g
else:
vals = self._M.values()
M = self._M
prec = ceil(max(log(M.order(), 2), 52) + 1) + 17
#print prec
RR = RealField(prec)
CC = ComplexField(prec)
d = self._d
m = self._m
if debug > 0: print d, m
if self._alpha3 == None:
if self._aniso_formula:
self._alpha4 = 1
self._alpha3 = -sum(
[BB(a) * mm for a, mm in self._v2.iteritems() if a != 0])
#print self._alpha3
self._alpha3 += Integer(d) - Integer(
1) - self._g.beta_formula()
#print self._alpha3, self._g.a5prime_formula()
self._alpha3 = self._alpha3 / RR(2)
else:
self._alpha3 = sum([(1 - a) * mm
for a, mm in self._v2.iteritems()
if a != 0])
#print self._alpha3
self._alpha3 += sum([(1 - a) * mm
for a, mm in vals.iteritems() if a != 0])
#print self._alpha3
self._alpha3 = self._alpha3 / Integer(2)
self._alpha4 = 1 / Integer(2) * (
vals[0] + v2) # the codimension of SkL in MkL
alpha3 = self._alpha3
alpha4 = self._alpha4
if debug > 0: print alpha3, alpha4
g1 = M.char_invariant(1)
g1 = CC(g1[0] * g1[1])
#print g1
g2 = M.char_invariant(2)
g2 = RR(real(g2[0] * g2[1]))
if debug > 0: print g2, g2.parent()
g3 = M.char_invariant(-3)
g3 = CC(g3[0] * g3[1])
if debug > 0:
print RR(d) / RR(4), sqrt(RR(m)) / RR(4), CC(
exp(2 * CC.pi() * CC(0, 1) * (2 * k + s) / Integer(8)))
alpha1 = RR(d) / RR(4) - (sqrt(RR(m)) / RR(4) * CC(
exp(2 * CC.pi() * CC(0, 1) * (2 * k + s) / Integer(8))) * g2)
if debug > 0: print alpha1
alpha2 = RR(d) / RR(3) + sqrt(RR(m)) / (3 * sqrt(RR(3))) * real(
exp(CC(2 * CC.pi() * CC(0, 1) * (4 * k + 3 * s - 10) / 24)) *
(g1 + g3))
if debug > 0: print alpha1, alpha2, g1, g2, g3, d, k, s
dim = real(d + (d * k / Integer(12)) - alpha1 - alpha2 - alpha3)
if debug > 0:
print "dimension:", dim
if abs(dim - round(dim)) > 1e-6:
raise RuntimeError(
"Error ({0}) too large in dimension formula for {1} and k={2}".
format(abs(dim - round(dim)),
self._M if self._M is not None else self._g, k))
dimr = dim
dim = Integer(round(dim))
if k >= 2 and dim < 0:
raise RuntimeError("Negative dimension (= {0}, {1})!".format(
dim, dimr))
return dim
def dimension(self, k, ignore=False, debug=0):
if k < 2 and not ignore:
raise NotImplementedError("k has to >= 2")
s = self._signature
if not (2 * k in ZZ):
raise ValueError("k has to be integral or half-integral")
if (2 * k + s) % 2 != 0:
return 0
m = self._m
n2 = self._n2
if self._v2.has_key(0):
v2 = self._v2[0]
else:
v2 = 1
if self._g != None:
if not self._aniso_formula:
vals = self._g.values()
#else:
#print "using aniso_formula"
M = self._g
else:
vals = self._M.values()
M = self._M
if (2 * k + s) % 4 == 0:
d = Integer(1) / Integer(2) * (
m + n2
) # |dimension of the Weil representation on even functions|
self._d = d
self._alpha4 = 1 / Integer(2) * (vals[0] + v2
) # the codimension of SkL in MkL
else:
d = Integer(1) / Integer(2) * (
m - n2
) # |dimension of the Weil representation on odd functions|
self._d = d
self._alpha4 = 1 / Integer(2) * (vals[0] - v2
) # the codimension of SkL in MkL
prec = ceil(max(log(M.order(), 2), 52) + 1) + 17
#print prec
RR = RealField(prec)
CC = ComplexField(prec)
if debug > 0: print "d, m = {0}, {1}".format(d, m)
eps = exp(2 * CC.pi() * CC(0, 1) * (s + 2 * k) / Integer(4))
eps = round(real(eps))
if self._alpha3 is None or self._last_eps != eps:
self._last_eps = eps
if self._aniso_formula:
self._alpha4 = 1
self._alpha3 = -sum(
[BB(a) * mm for a, mm in self._v2.iteritems() if a != 0])
#print self._alpha3
self._alpha3 += Integer(d) - Integer(
1) - self._g.beta_formula()
#print self._alpha3, self._g.a5prime_formula()
self._alpha3 = self._alpha3 / RR(2)
else:
self._alpha3 = eps * sum(
[(1 - a) * mm for a, mm in self._v2.iteritems() if a != 0])
if debug > 0: print "alpha3t = ", self._alpha3
self._alpha3 += sum([(1 - a) * mm
for a, mm in vals.iteritems() if a != 0])
#print self._alpha3
self._alpha3 = self._alpha3 / Integer(2)
alpha3 = self._alpha3
alpha4 = self._alpha4
if debug > 0: print alpha3, alpha4
g1 = M.char_invariant(1)
g1 = CC(g1[0] * g1[1])
#print g1
g2 = M.char_invariant(2)
g2 = CC(g2[0] * g2[1])
if debug > 0: print g2, g2.parent()
g3 = M.char_invariant(-3)
g3 = CC(g3[0] * g3[1])
if debug > 0: print "eps = {0}".format(eps)
if debug > 0:
print "d/4 = {0}, m/4 = {1}, e^(2pi i (2k+s)/8) = {2}".format(
RR(d) / RR(4),
sqrt(RR(m)) / RR(4),
CC(exp(2 * CC.pi() * CC(0, 1) * (2 * k + s) / Integer(8))))
if eps == 1:
g2_2 = real(g2)
else:
g2_2 = imag(g2) * CC(0, 1)
alpha1 = RR(d) / RR(4) - sqrt(RR(m)) / RR(4) * CC(
exp(2 * CC.pi() * CC(0, 1) * (2 * k + s) / Integer(8)) * g2_2)
if debug > 0: print alpha1
alpha2 = RR(d) / RR(3) + sqrt(RR(m)) / (3 * sqrt(RR(3))) * real(
exp(CC(2 * CC.pi() * CC(0, 1) * (4 * k + 3 * s - 10) / 24)) *
(g1 + eps * g3))
if debug > 0:
print "alpha1 = {0}, alpha2 = {1}, alpha3 = {2}, g1 = {3}, g2 = {4}, g3 = {5}, d = {6}, k = {7}, s = {8}".format(
alpha1, alpha2, alpha3, g1, g2, g3, d, k, s)
dim = real(d + (d * k / Integer(12)) - alpha1 - alpha2 - alpha3)
if debug > 0:
print "dimension:", dim
if abs(dim - round(dim)) > 1e-6:
raise RuntimeError(
"Error ({0}) too large in dimension formula for {1} and k={2}".
format(abs(dim - round(dim)),
self._M if self._M is not None else self._g, k))
dimr = dim
dim = Integer(round(dim))
if k >= 2 and dim < 0:
raise RuntimeError("Negative dimension (= {0}, {1})!".format(
dim, dimr))
return dim
def dimension(self,k,ignore=False, debug = 0):
if k < 2 and not ignore:
raise NotImplementedError("k has to >= 2")
s = self._signature
if not (2*k in ZZ):
raise ValueError("k has to be integral or half-integral")
if (2*k+s)%2 != 0:
return 0
m = self._m
n2 = self._n2
if self._v2.has_key(0):
v2 = self._v2[0]
else:
v2 = 1
if self._g != None:
if not self._aniso_formula:
vals = self._g.values()
#else:
#print "using aniso_formula"
M = self._g
else:
vals = self._M.values()
M = self._M
if (2*k+s)%4 == 0:
d = Integer(1)/Integer(2)*(m+n2) # |dimension of the Weil representation on even functions|
self._d = d
self._alpha4 = 1/Integer(2)*(vals[0]+v2) # the codimension of SkL in MkL
else:
d = Integer(1)/Integer(2)*(m-n2) # |dimension of the Weil representation on odd functions|
self._d = d
self._alpha4 = 1/Integer(2)*(vals[0]-v2) # the codimension of SkL in MkL
prec = ceil(max(log(M.order(),2),52)+1)+17
#print prec
RR = RealField(prec)
CC = ComplexField(prec)
if debug > 0: print "d, m = {0}, {1}".format(d,m)
eps = exp( 2 * CC.pi() * CC(0,1) * (s + 2*k) / Integer(4) )
eps = round(real(eps))
if self._alpha3 is None or self._last_eps != eps:
self._last_eps = eps
if self._aniso_formula:
self._alpha4 = 1
self._alpha3 = -sum([BB(a)*mm for a,mm in self._v2.iteritems() if a != 0])
#print self._alpha3
self._alpha3 += Integer(d) - Integer(1) - self._g.beta_formula()
#print self._alpha3, self._g.a5prime_formula()
self._alpha3 = self._alpha3/RR(2)
else:
self._alpha3 = eps*sum([(1-a)*mm for a,mm in self._v2.iteritems() if a != 0])
if debug>0: print "alpha3t = ", self._alpha3
self._alpha3 += sum([(1-a)*mm for a,mm in vals.iteritems() if a != 0])
#print self._alpha3
self._alpha3 = self._alpha3 / Integer(2)
alpha3 = self._alpha3
alpha4 = self._alpha4
if debug > 0: print alpha3, alpha4
g1=M.char_invariant(1)
g1=CC(g1[0]*g1[1])
#print g1
g2=M.char_invariant(2)
g2=CC(g2[0]*g2[1])
if debug > 0: print g2, g2.parent()
g3=M.char_invariant(-3)
g3=CC(g3[0]*g3[1])
if debug > 0: print "eps = {0}".format(eps)
if debug > 0: print "d/4 = {0}, m/4 = {1}, e^(2pi i (2k+s)/8) = {2}".format(RR(d) / RR(4), sqrt(RR(m)) / RR(4), CC(exp(2 * CC.pi() * CC(0,1) * (2 * k + s) / Integer(8))))
if eps == 1:
g2_2 = real(g2)
else:
g2_2 = imag(g2)*CC(0,1)
alpha1 = RR(d) / RR(4) - sqrt(RR(m)) / RR(4) * CC(exp(2 * CC.pi() * CC(0,1) * (2 * k + s) / Integer(8)) * g2_2)
if debug > 0: print alpha1
alpha2 = RR(d) / RR(3) + sqrt(RR(m)) / (3 * sqrt(RR(3))) * real(exp(CC(2 * CC.pi() * CC(0,1) * (4 * k + 3 * s - 10) / 24)) * (g1 + eps*g3))
if debug > 0: print "alpha1 = {0}, alpha2 = {1}, alpha3 = {2}, g1 = {3}, g2 = {4}, g3 = {5}, d = {6}, k = {7}, s = {8}".format(alpha1, alpha2, alpha3, g1, g2, g3, d, k, s)
dim = real(d + (d * k / Integer(12)) - alpha1 - alpha2 - alpha3)
if debug > 0:
print "dimension:", dim
if abs(dim-round(dim)) > 1e-6:
raise RuntimeError("Error ({0}) too large in dimension formula for {1} and k={2}".format(abs(dim-round(dim)), self._M if self._M is not None else self._g, k))
dimr = dim
dim = Integer(round(dim))
if k >=2 and dim < 0:
raise RuntimeError("Negative dimension (= {0}, {1})!".format(dim, dimr))
return dim
