def check_add_qexp(dim, min_det=1, max_det=None, fix=False):
count = 0
query = {}
query['dim'] = int(dim)
query['det'] = {'$gte': int(min_det)}
if max_det:
query['det']['$lte'] = int(max_det)
else:
max_det = "infinity"
lat_set = lat.find(query)
print(
"%s lattices to examine of dimension %s and determinant between %s and %s."
% (lat_set.count(), dim, min_det, max_det))
if lat_set.count() == 0:
return None
print("checking whether the q expansion is stored...")
for l in lat_set:
print("Testing lattice %s" % l['label'])
if l['theta_series'] == "":
print("q expansion NOT stored")
if fix:
M = l['gram']
exp = [
int(i) for i in gp("Vec(1+2*'x*Ser(qfrep(" +
str(gp(matrix(M))) + ",150,0)))")
]
lat.update({'theta_series': l['theta_series']},
{"$set": {
'theta_series': exp
}},
upsert=True)
print("Fixed lattice %s" % l['label'])
else:
print("q expansion stored")
def check_add_qexp(dim, min_det=1, max_det=None, fix=False):
count = 0
query = {}
query['dim'] = int(dim)
query['det'] = {'$gte' : int(min_det)}
if max_det:
query['det']['$lte'] = int(max_det)
else:
max_det = "infinity"
lat_set = lat.find(query)
print("%s lattices to examine of dimension %s and determinant between %s and %s."
% (lat_set.count(), dim, min_det, max_det))
if lat_set.count() == 0:
return None
print("checking whether the q expansion is stored...")
for l in lat_set:
print("Testing lattice %s" % l['label'])
if l['theta_series'] == "":
print("q expansion NOT stored")
if fix:
M=l['gram']
exp=[int(i) for i in gp("Vec(1+2*'x*Ser(qfrep("+str(gp(matrix(M)))+",150,0)))")]
lat.update({'theta_series': l['theta_series']}, {"$set": {'theta_series': exp}}, upsert=True)
print("Fixed lattice %s" % l['label'])
else:
print("q expansion stored")
def AJ1(R, f, P, i=1):
#if f.degree() == 5:
# return AJ1odd(R, f, P, i)
assert f.degree() == 6
J = gp("AJ1({},[{},{}],my_mesh,{})".format(str(f), str(P[0]), str(P[1]),
str(i)))
return vector(R, (J[1], J[2]))
def gpK(self):
if not self.haskey('gpK'):
Qx = PolynomialRing(QQ,'x')
# while [1] is a perfectly good basis for Z, gp seems to want []
basis = [Qx(el.replace('a','x')) for el in self.zk()] if self.degree() > 1 else []
k1 = gp( "nfinit([%s,%s])" % (str(self.poly()),str(basis)) )
self._data['gpK'] = k1
return self._data['gpK']
def gpK(self):
if not self.haskey('gpK'):
Qx = PolynomialRing(QQ,'x')
# while [1] is a perfectly good basis for Z, gp seems to want []
basis = [Qx(el.replace('a','x')) for el in self.zk()] if self.degree() > 1 else []
k1 = gp( "nfinit([%s,%s])" % (str(self.poly()),str(basis)) )
self._data['gpK'] = k1
return self._data['gpK']
def __init__(self, number_field, mod_ideal=1, mod_archimedean=None):
if mod_archimedean is None:
mod_archimedean = [0] * len(number_field.real_places())
mod_ideal = number_field.ideal(mod_ideal)
bnf = gp(number_field.pari_bnf())
# Use PARI to compute ray class group
bnr = bnf.bnrinit([mod_ideal, mod_archimedean], 1)
invariants = bnr[5][2] # bnr.clgp.cyc
invariants = tuple([Integer(x) for x in invariants])
names = tuple(["I%i" % i for i in range(len(invariants))])
generators = bnr[5][3] # bnr.gen = bnr.clgp[3]
generators = [number_field.ideal(pari(x)) for x in generators]
AbelianGroup_class.__init__(self, invariants, names)
self.__number_field = number_field
self.__bnr = bnr
self.__pari_mod = bnr[2][1]
self.__mod_ideal = mod_ideal
self.__mod_arch = mod_archimedean
self.__generators = generators
def __init__(self, number_field, mod_ideal = 1, mod_archimedean = None):
if mod_archimedean == None:
mod_archimedean = [0] * len(number_field.real_places())
mod_ideal = number_field.ideal( mod_ideal )
bnf = gp(number_field.pari_bnf())
# Use PARI to compute ray class group
bnr = bnf.bnrinit([mod_ideal, mod_archimedean],1)
invariants = bnr[5][2] # bnr.clgp.cyc
invariants = tuple([ Integer(x) for x in invariants ])
names = tuple([ "I%i"%i for i in range(len(invariants)) ])
generators = bnr[5][3] # bnr.gen = bnr.clgp[3]
generators = [ number_field.ideal(pari(x)) for x in generators ]
AbelianGroup_class.__init__(self, invariants, names)
self.__number_field = number_field
self.__bnr = bnr
self.__pari_mod = bnr[2][1]
self.__mod_ideal = mod_ideal
self.__mod_arch = mod_archimedean
self.__generators = generators
def myZZ(val):
return int(ZZ(gp(val)))
def myZZ(val):
return int(ZZ(gp(val)))
def load_gp():
if not all([
gp("RNEG") == 101,
gp("RPOS") == 102,
gp("INEG") == 103,
gp("IPOS") == 104
]):
#print "loading gp code"
gp._reset_expect()
# Branch values:
gp("RNEG=101; RPOS=102; INEG=103; IPOS=104;")
# Print branch label given a branch value.
# The label corresponds to the branch cut.
gp("""
printbranch(branch)=
{
if( branch == RNEG, print("RNEG"));
if( branch == RPOS, print("RPOS"));
if( branch == INEG, print("INEG"));
if( branch == IPOS, print("IPOS"));
return();
error("Unknown sqrt branch");
}
""".replace('\n', ''))
# Square root determined by branch value, which should be thought of as a
# continuation to a half-plane.
gp("""
rac2(z, branch)=
{
if( branch == RNEG, return( sqrt(z) ) );
if( branch == RPOS, return( I * sqrt(-z) ) );
if( branch == INEG, return( sqrt(I) * sqrt(-I*z) ) );
if( branch == IPOS, return( sqrt(-I) * sqrt(I * z) ) );
error("Unknown sqrt branch");
}
""".replace('\n', ''))
# Best branch is the one that stays farthest away from the cut corresponding to
# it.
gp("""
best_branch(z)=
{
my( x = real(z), y = imag(z) );
if( abs(y) > abs(x),
if( y > 0, return(INEG), return(IPOS))
,
if( x > 0, return(RNEG), return(RPOS))
);
}
""".replace('\n', ''))
# Evaluation:
gp("EvPol(f, x) = subst(f, variable(f), x);")
# Heuristic zero test:
gp("IsApprox0(x) = abs(x) < 10^(-default(realprecision) / 2);")
# Return real positive roots of a complex polynomial
gp("""
{
polposroots( f ) =
if( poldegree( f ) <= 0, return( [] ) );
my( Z = polroots( f ) );
RZ = List();
for(i = 1,#Z,
if( IsApprox0( imag( Z[i] ) ) && real( Z[i] ) > 10^(-default(realprecision) / 2), listput( RZ, real(Z[i]) ))
);
Vec(RZ);
}
""".replace('\n', ''))
# Return distinct roots in [0, 1] of a complex polynomial
gp("""
{
pol01roots( f ) =
if( poldegree( f ) <= 0, return([]));
my( Z = polroots(f), i, z, eps = 10^(-default(realprecision) / 2));
RZ = List();
for(i = 1,#Z,
if( IsApprox0( imag(Z[i]) ),
z = real( Z[i] );
if( z > eps && z < 1-eps && (#RZ==0 || abs( z - RZ[#RZ]) > eps),
listput(RZ, z)
)
)
);
Vec(RZ);
}
""".replace('\n', ''))
# Normalises a polynomial by removing the null leading terms (should fix the bug reported by Jeroen)
gp("""
{
normalise_lc_pol(f) =
my( d = poldegree(f), d2 = -1, var = variable(f), eps=10^(-default(realprecision)/2) );
for(i=0,d,
if(abs(polcoeff(f,i))>eps,d2=i)
);
sum(i=0,d2,polcoeff(f,i)*var^i);
}
""".replace('\n', ''))
# Returns values in [0, 1] where f crosses one of the branch cuts
gp("""
{
polcrossaxes01(f) =
my( u = real(f), v = imag(f), Z, Zf, eps = 10^(-default(realprecision)/2) );
Z = pol01roots(f);
u = normalise_lc_pol(u);
v = normalise_lc_pol(v);
if( u == 0 || v == 0, return(Z));
Zf=prod(i=1,#Z,'t-Z[i]);
Z=concat(Z,pol01roots(u\Zf)); Z=concat(Z,pol01roots(v\Zf));
Z=vecsort(Z); /* TODO fix can be multiple roots there */
}
""".replace('\n', ''))
# Integrates division by sqrt(f) from a to b (complexes), assuming no branch
# cut and f square free. Detects zeros at ends and simplifies them out by changing
# variable.
gp("""{
AJ_fin_nocut(f, a, b, branch, mesh)=
/*print("AJfin");*/
my(t, g, h, S, z0, z1);
g = EvPol(f, a + (b - a) * 't);
z0 = IsApprox0( polcoeff(g, 0) );
z1 = IsApprox0( EvPol(g, 1) );
if(z0,
if(z1,
/*print("we have zeros at both ends");*/
h = g\('t-'t^2);
S = 6 * intnum( t = 0, 1, [1, a + (b - a) * t^2 * (3 - 2*t)] / rac2((3 - 2*t)*(2*t + 1) * EvPol(h, t^2 * (3 - 2*t)) ,branch),mesh)
,
/*print("a is a zero, and b is not");*/
h = g\ 't;
S = 2 * intnum( t = 0, 1, [1 , a + (b - a) * t^2 ] / rac2( EvPol( h, t^2 ), branch), mesh)
)
,
if(z1,
/*print("b is zero, and a is not");*/
h = g\(1-'t);
S = 2 * intnum(t = 0, 1, [1, b + (a - b) * t^2 ] / rac2( EvPol( h, 1 - t^2 ), branch), mesh)
,
/*print("neither a or b are zeros");*/
S = intnum(t = 0, 1, [1, a + (b - a) * t ] / rac2(EvPol(g, t), branch), mesh)
)
);
(b - a) * S;
}""".replace('\n', ''))
# Abelian integral from x=a straight to x=b, starting at y=ya. Also returns yb.
# Technique: Keep track of a branch switch throughout.
# (The correct signed root is chosen by explicit continuation.)
gp("""
AJ_fin(f, a, b, ya, mesh) =
{
my(g, stops, S = [0, 0], s, yprev, z, i, w, branch, ynew);
/*print(["Int",a,b]);*/
g = EvPol(f, a + (b - a) * 't);
/* Checks for which 0<t<1 the image by f of the integration path crosses the axes */
stops = polcrossaxes01( g );
/*print(Str(#stops, "obstacles"));*/
/* Turn these t into z */
stops = apply( t-> a + t * (b-a), stops);
/* Add endpoints */
stops = concat([a], concat(stops,[b]));
/* print(stops); */
/* Keep track of the sheet */
s = 1;
yprev = ya;
z = a;
for(i = 1,#stops - 1,
w = EvPol(f,z);
/* w is in the area we're starting from now. */
if( IsApprox0(w), w = EvPol(deriv(f),z));
branch = best_branch(w);
/* Choose a sqrt accordingly */
/* print(Str("New value ",w)); */
/* printbranch(branch); */
if( !IsApprox0( EvPol(f, z) ),
/* adjust sign so that we stay in the same sheet */
ynew = rac2( EvPol(f, z), branch);
if( s * real(ynew/yprev) < 0, s = -s );
);
S += s * AJ_fin_nocut(f, stops[i], stops[i+1], branch, mesh);
/* integrate 1 step */
/* compute new start and new y */
z = stops[i + 1];
yprev = s * rac2(EvPol(f, z) , branch);
);
[S, yprev];
}
""".replace('\n', ''))
# Image of P+Q-OO by Abel-Jacobi
gp("""
{
AJ2(f, P, Q, mesh) =
my();
Z = polroots(f);
/* Integrate P+Q-2(z,0) instead, it is the same up to a period */
z = Z[1];
AJ_fin(f, P[1], z, P[2], mesh)[1] + AJ_fin(f, Q[1], z, Q[2], mesh)[1];
}
""".replace('\n', ''))
# Image of P-OO by Abel-Jacobi
gp("""
{
AJ1(f,P,mesh)=
AJ1(f, P, mesh, 1);
}
""".replace('\n', ''))
gp("""
{
AJ1(f,P,mesh,i)=
my();
Z = polroots(f);
z = Z[i]; /* Integrate P-(z,0) instead, it is the same up to a period */
AJ_fin(f,P[1],z,P[2],mesh)[1];
}
""".replace('\n', ''))
gp("""
{
whatZ(f)=
Z = polroots(f);
z = Z[1];
z;
}
""".replace('\n', ''))
def Z(f):
print gp("whatZ({})".format(str(f)))
def AJ2(f, P, Q):
J = gp("AJ2({},[{},{}],[{},{}],my_mesh)".format(str(f), str(P[0]),
str(P[1]), str(Q[0]),
str(Q[1])))
return (J[1], J[2])
def PariIntNumInit(n=1):
gp("my_mesh = intnuminit(0,1,{})".format(str(n)))