def enumerate_totallyreal_fields_all(n, B, verbose=0, return_seqs=False):
r"""
Enumerates *all* totally real fields of degree `n` with discriminant `\le B`,
primitive or otherwise.
EXAMPLES::
sage: enumerate_totallyreal_fields_all(4, 2000)
[[725, x^4 - x^3 - 3*x^2 + x + 1],
[1125, x^4 - x^3 - 4*x^2 + 4*x + 1],
[1600, x^4 - 6*x^2 + 4],
[1957, x^4 - 4*x^2 - x + 1],
[2000, x^4 - 5*x^2 + 5]]
In practice most of these will be found by :func:`~sage.rings.number_field.totallyreal.enumerate_totallyreal_fields_prim`, which is guaranteed to return all primitive fields but often returns many non-primitive ones as well. For instance, only one of the five fields in the example above is primitive, but :func:`~sage.rings.number_field.totallyreal.enumerate_totallyreal_fields_prim` finds four out of the five (the exception being `x^4 - 6x^2 + 4`).
TESTS:
The following was fixed in :trac:`13101`::
sage: enumerate_totallyreal_fields_all(8, 10^6) # long time (about 2 s)
[]
"""
S = []
counts = [0,0,0]
if len(divisors(n)) > 4:
raise ValueError, "Only implemented for n = p*q with p,q prime"
for d in divisors(n):
if d > 1 and d < n:
Sds = enumerate_totallyreal_fields_prim(d, int(math.floor((1.*B)**(1.*d/n))), verbose=verbose)
for i in range(len(Sds)):
if verbose:
print "="*80
print "Taking F =", Sds[i][1]
F = NumberField(ZZx(Sds[i][1]), 't')
T = enumerate_totallyreal_fields_rel(F, n/d, B, verbose=verbose, return_seqs=return_seqs)
if return_seqs:
for i in range(3):
counts[i] += T[0][i]
S += [[t[0],pari(t[1]).Polrev()] for t in T[1]]
else:
S += [[t[0],t[1]] for t in T]
j = i+1
for E in enumerate_totallyreal_fields_prim(n/d, int(math.floor((1.*B)**(1./d)/(1.*Sds[i][0])**(n*1./d**2)))):
for EF in F.composite_fields(NumberField(ZZx(E[1]), 'u')):
if EF.degree() == n and EF.disc() <= B:
S.append([EF.disc(), pari(EF.absolute_polynomial())])
S += enumerate_totallyreal_fields_prim(n, B, verbose=verbose)
S.sort()
weed_fields(S)
# Output.
if verbose:
saveout = sys.stdout
if type(verbose) == str:
fsock = open(verbose, 'w')
sys.stdout = fsock
# Else, print to screen
print "="*80
print "Polynomials tested:", counts[0]
print "Irreducible polynomials:", counts[1]
print "Polynomials with nfdisc <= B:", counts[2]
for i in range(len(S)):
print S[i]
if type(verbose) == str:
fsock.close()
sys.stdout = saveout
if return_seqs:
return [counts,[[s[0],s[1].reverse().Vec()] for s in S]]
else:
return S
def enumerate_totallyreal_fields_all(n, B, verbose=0, return_seqs=False,
return_pari_objects=True):
r"""
Enumerates *all* totally real fields of degree ``n`` with discriminant
at most ``B``, primitive or otherwise.
INPUT:
- ``n`` -- integer, the degree
- ``B`` -- integer, the discriminant bound
- ``verbose`` -- boolean or nonnegative integer or string (default: 0)
give a verbose description of the computations being performed. If
``verbose`` is set to ``2`` or more then it outputs some extra
information. If ``verbose`` is a string then it outputs to a file
specified by ``verbose``
- ``return_seqs`` -- (boolean, default False) If ``True``, then return
the polynomials as sequences (for easier exporting to a file). This
also returns a list of four numbers, as explained in the OUTPUT
section below.
- ``return_pari_objects`` -- (boolean, default: True) if both
``return_seqs`` and ``return_pari_objects`` are ``False`` then it
returns the elements as Sage objects; otherwise it returns pari
objects.
EXAMPLES::
sage: enumerate_totallyreal_fields_all(4, 2000)
[[725, x^4 - x^3 - 3*x^2 + x + 1],
[1125, x^4 - x^3 - 4*x^2 + 4*x + 1],
[1600, x^4 - 6*x^2 + 4],
[1957, x^4 - 4*x^2 - x + 1],
[2000, x^4 - 5*x^2 + 5]]
sage: enumerate_totallyreal_fields_all(1, 10)
[[1, x - 1]]
TESTS:
Each of the outputs must be elements of Sage if ``return_pari_objects``
is set to ``False``::
sage: enumerate_totallyreal_fields_all(2, 10)
[[5, x^2 - x - 1], [8, x^2 - 2]]
sage: enumerate_totallyreal_fields_all(2, 10)[0][1].parent()
Interface to the PARI C library
sage: enumerate_totallyreal_fields_all(2, 10, return_pari_objects=False)[0][1].parent()
Univariate Polynomial Ring in x over Rational Field
In practice most of these will be found by
:func:`~sage.rings.number_field.totallyreal.enumerate_totallyreal_fields_prim`,
which is guaranteed to return all primitive fields but often returns
many non-primitive ones as well. For instance, only one of the five
fields in the example above is primitive, but
:func:`~sage.rings.number_field.totallyreal.enumerate_totallyreal_fields_prim`
finds four out of the five (the exception being `x^4 - 6x^2 + 4`).
The following was fixed in :trac:`13101`::
sage: enumerate_totallyreal_fields_all(8, 10^6) # long time (about 2 s)
[]
"""
S = []
counts = [0,0,0,0]
if len(divisors(n)) > 4:
raise ValueError("Only implemented for n = p*q with p,q prime")
for d in divisors(n):
if d > 1 and d < n:
Sds = enumerate_totallyreal_fields_prim(d, int(math.floor((1.*B)**(1.*d/n))), verbose=verbose)
for i in range(len(Sds)):
if verbose:
print "="*80
print "Taking F =", Sds[i][1]
F = NumberField(ZZx(Sds[i][1]), 't')
T = enumerate_totallyreal_fields_rel(F, n/d, B, verbose=verbose, return_seqs=return_seqs)
if return_seqs:
for i in range(4):
counts[i] += T[0][i]
S += [[t[0],pari(t[1]).Polrev()] for t in T[1]]
else:
S += [[t[0],t[1]] for t in T]
j = i+1
for E in enumerate_totallyreal_fields_prim(n/d, int(math.floor((1.*B)**(1./d)/(1.*Sds[i][0])**(n*1./d**2)))):
for EF in F.composite_fields(NumberField(ZZx(E[1]), 'u')):
if EF.degree() == n and EF.disc() <= B:
S.append([EF.disc(), pari(EF.absolute_polynomial())])
S += enumerate_totallyreal_fields_prim(n, B, verbose=verbose)
S.sort(cmp=lambda x, y: cmp(x[0], y[0]) or cmp(x[1], y[1]))
weed_fields(S)
# Output.
if verbose:
saveout = sys.stdout
if isinstance(verbose, str):
fsock = open(verbose, 'w')
sys.stdout = fsock
# Else, print to screen
print "="*80
print "Polynomials tested: {}".format(counts[0])
print ( "Polynomials with discriminant with large enough square"
" divisor: {}".format(counts[1]))
print "Irreducible polynomials: {}".format(counts[2])
print "Polynomials with nfdisc <= B: {}".format(counts[3])
for i in range(len(S)):
print S[i]
if isinstance(verbose, str):
fsock.close()
sys.stdout = saveout
# Make sure to return elements that belong to Sage
if return_seqs:
return [[ZZ(_) for _ in counts],
[[ZZ(s[0]), [QQ(_) for _ in s[1].polrecip().Vec()]] for s in S]]
elif return_pari_objects:
return S
else:
Px = PolynomialRing(QQ, 'x')
return [[ZZ(s[0]), Px([QQ(_) for _ in s[1].list()])]
for s in S]
def enumerate_totallyreal_fields_rel(F, m, B, a = [], verbose=0, return_seqs=False):
r"""
This function enumerates (primitive) totally real field extensions of
degree `m>1` of the totally real field F with discriminant `d \leq B`;
optionally one can specify the first few coefficients, where the sequence ``a``
corresponds to a polynomial by
::
a[d]*x^n + ... + a[0]*x^(n-d)
if ``length(a) = d+1``, so in particular always ``a[d] = 1``.
If verbose == 1 (or 2), then print to the screen (really) verbosely; if
verbose is a string, then print verbosely to the file specified by verbose.
If return_seqs, then return the polynomials as sequences (for easier
exporting to a file).
NOTE:
This is guaranteed to give all primitive such fields, and
seems in practice to give many imprimitive ones.
INPUT:
- ``F`` -- number field, the base field
- ``m`` -- integer, the degree
- ``B`` -- integer, the discriminant bound
- ``a`` -- list (default: []), the coefficient list to begin with
- ``verbose`` -- boolean or string (default: 0)
- ``return_seqs`` -- boolean (default: False)
OUTPUT:
the list of fields with entries [d,fabs,f], where
d is the discriminant, fabs is an absolute defining polynomial,
and f is a defining polynomial relative to F,
sorted by discriminant.
EXAMPLES::
sage: ZZx = ZZ['x']
sage: F.<t> = NumberField(x^2-2)
sage: enumerate_totallyreal_fields_rel(F, 2, 2000)
[[1600, x^4 - 6*x^2 + 4, xF^2 + xF - 1]]
AUTHORS:
- John Voight (2007-11-01)
"""
if not isinstance(m, Integer):
try:
m = Integer(m)
except TypeError:
raise TypeError, "cannot coerce m (= %s) to an integer" % m
if (m < 1):
raise ValueError, "m must be at least 1."
n = F.degree()*m
# Initialize
T = tr_data_rel(F,m,B,a)
S = []
Srel = []
dB_odlyzko = odlyzko_bound_totallyreal(n)
dB = math.ceil(40000*dB_odlyzko**n)
counts = [0,0,0,0]
# Trivial case
if m == 1:
g = pari(F.defining_polynomial()).reverse().Vec()
if return_seqs:
return [[0,0,0,0],[1,g,[-1,1]]]
else:
return [[1,pari('x-1'),g]]
if verbose:
saveout = sys.stdout
if type(verbose) == str:
fsock = open(verbose, 'w')
sys.stdout = fsock
# Else, print to screen
f_out = [0]*m + [1]
if verbose == 2:
T.incr(f_out,verbose)
else:
T.incr(f_out)
Fx = PolynomialRing(F, 'xF')
nfF = pari(str(F.defining_polynomial()).replace('x', str(F.primitive_element()) ) )
parit = pari(str(F.primitive_element()))
while f_out[m] != 0:
counts[0] += 1
if verbose:
print "==>", f_out,
f_str = ''
for i in range(len(f_out)):
f_str += '(' + str(f_out[i]) + ')*x^' + str(i)
if i < len(f_out)-1:
f_str += '+'
nf = pari(f_str)
if nf.poldegree('t') == 0:
nf = nf.subst('x', 'x-t')
nf = nf.polresultant(nfF, parit)
d = nf.poldisc()
counts[0] += 1
if d > 0 and nf.polsturm_full() == n:
da = int_has_small_square_divisor(Integer(d))
if d > dB or d <= B*da:
counts[1] += 1
if nf.polisirreducible():
counts[2] += 1
[zk,d] = nf.nfbasis_d()
if d <= B:
if verbose:
print "has discriminant", d,
# Find a minimal lattice element
counts[3] += 1
ng = pari([nf,zk]).polredabs()
# Check if K is contained in the list.
found = False
ind = bisect.bisect_left(S, [d,ng])
while ind < len(S) and S[ind][0] == d:
if S[ind][1] == ng:
if verbose:
print "but is not new"
found = True
break
ind += 1
if not found:
if verbose:
print "and is new!"
S.insert(ind, [d,ng])
Srel.insert(ind, Fx(f_out))
else:
if verbose:
print "has discriminant", abs(d), "> B"
else:
if verbose:
print "is not absolutely irreducible"
else:
if verbose:
print "has discriminant", abs(d), "with no large enough square divisor"
else:
if verbose:
if d == 0:
print "is not squarefree"
else:
print "is not totally real"
if verbose == 2:
T.incr(f_out,verbose=verbose)
else:
T.incr(f_out)
# In the application of Smyth's theorem above, we exclude finitely
# many possibilities which we must now throw back in.
if m == 2:
if Fx([-1,1,1]).is_irreducible():
K = F.extension(Fx([-1,1,1]), 'tK')
Kabs = K.absolute_field('tKabs')
Kabs_pari = pari(Kabs.defining_polynomial())
d = K.absolute_discriminant()
if abs(d) <= B:
ng = Kabs_pari.polredabs()
ind = bisect.bisect_left(S, [d,ng])
S.insert(ind, [d,ng])
Srel.insert(ind, Fx([-1,1,1]))
elif F.degree() == 2:
for ff in [[1,-7,13,-7,1],[1,-8,14,-7,1]]:
f = Fx(ff).factor()[0][0]
K = F.extension(f, 'tK')
Kabs = K.absolute_field('tKabs')
Kabs_pari = pari(Kabs.defining_polynomial())
d = K.absolute_discriminant()
if abs(d) <= B:
ng = Kabs_pari.polredabs()
ind = bisect.bisect_left(S, [d,ng])
S.insert(ind, [d,ng])
Srel.insert(ind, f)
elif m == 3:
if Fx([-1,6,-5,1]).is_irreducible():
K = F.extension(Fx([-1,6,-5,1]), 'tK')
Kabs = K.absolute_field('tKabs')
Kabs_pari = pari(Kabs.defining_polynomial())
d = K.absolute_discriminant()
if abs(d) <= B:
ng = Kabs_pari.polredabs()
ind = bisect.bisect_left(S, [d,ng])
S.insert(ind, [d,ng])
Srel.insert(ind, Fx([-1,6,-5,1]))
# Now check for isomorphic fields
S = [[S[i][0],S[i][1],Srel[i]] for i in range(len(S))]
weed_fields(S)
# Output.
if verbose:
print "="*80
print "Polynomials tested:", counts[0]
print "Irreducible polynomials:", counts[1]
print "Polynomials with nfdisc <= B:", counts[2]
for i in range(len(S)):
print S[i]
if type(verbose) == str:
fsock.close()
sys.stdout = saveout
if return_seqs:
return [counts,[[s[0],s[1].reverse().Vec(),s[2].coeffs()] for s in S]]
else:
return S
def enumerate_totallyreal_fields_rel(F, m, B, a = [], verbose=0,
return_seqs=False,
return_pari_objects=True):
r"""
This function enumerates (primitive) totally real field extensions of
degree `m>1` of the totally real field F with discriminant `d \leq B`;
optionally one can specify the first few coefficients, where the sequence ``a``
corresponds to a polynomial by
::
a[d]*x^n + ... + a[0]*x^(n-d)
if ``length(a) = d+1``, so in particular always ``a[d] = 1``.
.. note::
This is guaranteed to give all primitive such fields, and
seems in practice to give many imprimitive ones.
INPUT:
- ``F`` -- number field, the base field
- ``m`` -- integer, the degree
- ``B`` -- integer, the discriminant bound
- ``a`` -- list (default: []), the coefficient list to begin with
- ``verbose`` -- boolean or nonnegative integer or string (default: 0)
give a verbose description of the computations being performed. If
``verbose`` is set to ``2`` or more then it outputs some extra
information. If ``verbose`` is a string then it outputs to a file
specified by ``verbose``
- ``return_seqs`` -- (boolean, default False) If ``True``, then return
the polynomials as sequences (for easier exporting to a file). This
also returns a list of four numbers, as explained in the OUTPUT
section below.
- ``return_pari_objects`` -- (boolean, default: True) if
both ``return_seqs`` and ``return_pari_objects`` are ``False`` then
it returns the elements as Sage objects; otherwise it returns pari
objects.
OUTPUT:
- the list of fields with entries ``[d,fabs,f]``, where ``d`` is the
discriminant, ``fabs`` is an absolute defining polynomial, and ``f``
is a defining polynomial relative to ``F``, sorted by discriminant.
- if ``return_seqs`` is ``True``, then the first field of the list is
a list containing the count of four items as explained below
- the first entry gives the number of polynomials tested
- the second entry gives the number of polynomials with its
discriminant having a large enough square divisor
- the third entry is the number of irreducible polynomials
- the fourth entry is the number of irreducible polynomials with
discriminant at most ``B``
EXAMPLES::
sage: ZZx = ZZ['x']
sage: F.<t> = NumberField(x^2-2)
sage: enumerate_totallyreal_fields_rel(F, 1, 2000)
[[1, [-2, 0, 1], xF - 1]]
sage: enumerate_totallyreal_fields_rel(F, 2, 2000)
[[1600, x^4 - 6*x^2 + 4, xF^2 + xF - 1]]
sage: enumerate_totallyreal_fields_rel(F, 2, 2000, return_seqs=True)
[[9, 6, 5, 0], [[1600, [4, 0, -6, 0, 1], [-1, 1, 1]]]]
TESTS:
Each of the outputs must be elements of Sage if ``return_pari_objects``
is set to ``False``::
sage: enumerate_totallyreal_fields_rel(F, 2, 2000)[0][1].parent()
Interface to the PARI C library
sage: enumerate_totallyreal_fields_rel(F, 2, 2000, return_pari_objects=False)[0][0].parent()
Integer Ring
sage: enumerate_totallyreal_fields_rel(F, 2, 2000, return_pari_objects=False)[0][1].parent()
Univariate Polynomial Ring in x over Rational Field
sage: enumerate_totallyreal_fields_rel(F, 2, 2000, return_pari_objects=False)[0][2].parent()
Univariate Polynomial Ring in xF over Number Field in t with defining polynomial x^2 - 2
sage: enumerate_totallyreal_fields_rel(F, 2, 2000, return_seqs=True)[1][0][1][0].parent()
Rational Field
AUTHORS:
- John Voight (2007-11-01)
"""
if not isinstance(m, Integer):
try:
m = Integer(m)
except TypeError:
raise TypeError("cannot coerce m (= %s) to an integer" % m)
if (m < 1):
raise ValueError("m must be at least 1.")
n = F.degree()*m
# Initialize
S = {} # dictionary of the form {(d, fabs): f, ...}
dB_odlyzko = odlyzko_bound_totallyreal(n)
dB = math.ceil(40000*dB_odlyzko**n)
counts = [0,0,0,0]
# Trivial case
if m == 1:
g = pari(F.defining_polynomial()).polrecip().Vec()
if return_seqs:
return [[0,0,0,0], [1, [-1, 1], g]]
elif return_pari_objects:
return [[1, g, pari('xF-1')]]
else:
Px = PolynomialRing(QQ, 'xF')
return [[ZZ(1), [QQ(_) for _ in g], Px.gen()-1]]
if verbose:
saveout = sys.stdout
if isinstance(verbose, str):
fsock = open(verbose, 'w')
sys.stdout = fsock
# Else, print to screen
f_out = [0]*m + [1]
T = tr_data_rel(F,m,B,a)
if verbose == 2:
T.incr(f_out,verbose)
else:
T.incr(f_out)
Fx = PolynomialRing(F, 'xF')
nfF = pari(str(F.defining_polynomial()).replace('x', str(F.primitive_element()) ) )
parit = pari(str(F.primitive_element()))
while f_out[m] != 0:
counts[0] += 1
if verbose:
print "==>", f_out,
f_str = ''
for i in range(len(f_out)):
f_str += '(' + str(f_out[i]) + ')*x^' + str(i)
if i < len(f_out)-1:
f_str += '+'
nf = pari(f_str)
if nf.poldegree('t') == 0:
nf = nf.subst('x', 'x-t')
nf = nf.polresultant(nfF, parit)
d = nf.poldisc()
#counts[0] += 1
if d > 0 and nf.polsturm() == n:
da = int_has_small_square_divisor(Integer(d))
if d > dB or d <= B*da:
counts[1] += 1
if nf.polisirreducible():
counts[2] += 1
[zk,d] = nf.nfbasis_d()
if d <= B:
if verbose:
print "has discriminant", d,
# Find a minimal lattice element
counts[3] += 1
ng = pari([nf,zk]).polredabs()
# Check if K is contained in the list.
if (d, ng) in S:
if verbose:
print "but is not new"
else:
if verbose:
print "and is new!"
S[(d, ng)] = Fx(f_out)
else:
if verbose:
print "has discriminant", abs(d), "> B"
else:
if verbose:
print "is not absolutely irreducible"
else:
if verbose:
print "has discriminant", abs(d), "with no large enough square divisor"
else:
if verbose:
if d == 0:
print "is not squarefree"
else:
print "is not totally real"
if verbose == 2:
T.incr(f_out,verbose=verbose)
else:
T.incr(f_out)
# In the application of Smyth's theorem above, we exclude finitely
# many possibilities which we must now throw back in.
if m == 2:
if Fx([-1,1,1]).is_irreducible():
K = F.extension(Fx([-1,1,1]), 'tK')
Kabs = K.absolute_field('tKabs')
Kabs_pari = pari(Kabs.defining_polynomial())
d = K.absolute_discriminant()
if abs(d) <= B:
ng = Kabs_pari.polredabs()
S[(d, ng)] = Fx([-1,1,1])
elif F.degree() == 2:
for ff in [[1,-7,13,-7,1],[1,-8,14,-7,1]]:
f = Fx(ff).factor()[0][0]
K = F.extension(f, 'tK')
Kabs = K.absolute_field('tKabs')
Kabs_pari = pari(Kabs.defining_polynomial())
d = K.absolute_discriminant()
if abs(d) <= B:
ng = Kabs_pari.polredabs()
S[(d, ng)] = f
elif m == 3:
if Fx([-1,6,-5,1]).is_irreducible():
K = F.extension(Fx([-1,6,-5,1]), 'tK')
Kabs = K.absolute_field('tKabs')
Kabs_pari = pari(Kabs.defining_polynomial())
d = K.absolute_discriminant()
if abs(d) <= B:
ng = Kabs_pari.polredabs()
S[(d, ng)] = Fx([-1,6,-5,1])
# Convert S to a sorted list of triples [d, fabs, f], taking care
# to use cmp() and not the comparison operators on PARI polynomials.
S = [[s[0], s[1], t] for s, t in S.items()]
S.sort(cmp=lambda x, y: cmp(x[0], y[0]) or cmp(x[1], y[1]))
# Now check for isomorphic fields
weed_fields(S)
# Output.
if verbose:
print "="*80
print "Polynomials tested: {}".format(counts[0])
print ( "Polynomials with discriminant with large enough square"
" divisor: {}".format(counts[1]))
print "Irreducible polynomials: {}".format(counts[2])
print "Polynomials with nfdisc <= B: {}".format(counts[3])
for i in range(len(S)):
print S[i]
if isinstance(verbose, str):
fsock.close()
sys.stdout = saveout
# Make sure to return elements that belong to Sage
if return_seqs:
return [[ZZ(x) for x in counts],
[[s[0], [QQ(x) for x in s[1].polrecip().Vec()], s[2].coefficients(sparse=False)]
for s in S]
]
elif return_pari_objects:
return S
else:
Px = PolynomialRing(QQ, 'x')
return [[s[0], Px([QQ(_) for _ in s[1].list()]), s[2]] for s in S]
def enumerate_totallyreal_fields_all(n,
B,
verbose=0,
return_seqs=False,
return_pari_objects=True):
r"""
Enumerates *all* totally real fields of degree ``n`` with discriminant
at most ``B``, primitive or otherwise.
INPUT:
- ``n`` -- integer, the degree
- ``B`` -- integer, the discriminant bound
- ``verbose`` -- boolean or nonnegative integer or string (default: 0)
give a verbose description of the computations being performed. If
``verbose`` is set to ``2`` or more then it outputs some extra
information. If ``verbose`` is a string then it outputs to a file
specified by ``verbose``
- ``return_seqs`` -- (boolean, default False) If ``True``, then return
the polynomials as sequences (for easier exporting to a file). This
also returns a list of four numbers, as explained in the OUTPUT
section below.
- ``return_pari_objects`` -- (boolean, default: True) if both
``return_seqs`` and ``return_pari_objects`` are ``False`` then it
returns the elements as Sage objects; otherwise it returns pari
objects.
EXAMPLES::
sage: enumerate_totallyreal_fields_all(4, 2000)
[[725, x^4 - x^3 - 3*x^2 + x + 1],
[1125, x^4 - x^3 - 4*x^2 + 4*x + 1],
[1600, x^4 - 6*x^2 + 4],
[1957, x^4 - 4*x^2 - x + 1],
[2000, x^4 - 5*x^2 + 5]]
sage: enumerate_totallyreal_fields_all(1, 10)
[[1, x - 1]]
TESTS:
Each of the outputs must be elements of Sage if ``return_pari_objects``
is set to ``False``::
sage: enumerate_totallyreal_fields_all(2, 10)
[[5, x^2 - x - 1], [8, x^2 - 2]]
sage: type(enumerate_totallyreal_fields_all(2, 10)[0][1])
<type 'cypari2.gen.Gen'>
sage: enumerate_totallyreal_fields_all(2, 10, return_pari_objects=False)[0][1].parent()
Univariate Polynomial Ring in x over Rational Field
In practice most of these will be found by
:func:`~sage.rings.number_field.totallyreal.enumerate_totallyreal_fields_prim`,
which is guaranteed to return all primitive fields but often returns
many non-primitive ones as well. For instance, only one of the five
fields in the example above is primitive, but
:func:`~sage.rings.number_field.totallyreal.enumerate_totallyreal_fields_prim`
finds four out of the five (the exception being `x^4 - 6x^2 + 4`).
The following was fixed in :trac:`13101`::
sage: enumerate_totallyreal_fields_all(8, 10^6) # long time (about 2 s)
[]
"""
S = []
counts = [0, 0, 0, 0]
if len(divisors(n)) > 4:
raise ValueError("Only implemented for n = p*q with p,q prime")
for d in divisors(n):
if d > 1 and d < n:
Sds = enumerate_totallyreal_fields_prim(
d, int(math.floor((1. * B)**(1. * d / n))), verbose=verbose)
for i in range(len(Sds)):
if verbose:
print("=" * 80)
print("Taking F =", Sds[i][1])
F = NumberField(ZZx(Sds[i][1]), 't')
T = enumerate_totallyreal_fields_rel(F,
n / d,
B,
verbose=verbose,
return_seqs=return_seqs)
if return_seqs:
for i in range(4):
counts[i] += T[0][i]
S += [[t[0], pari(t[1]).Polrev()] for t in T[1]]
else:
S += [[t[0], t[1]] for t in T]
for E in enumerate_totallyreal_fields_prim(
n / d,
int(
math.floor((1. * B)**(1. / d) /
(1. * Sds[i][0])**(n * 1. / d**2)))):
for EF in F.composite_fields(NumberField(ZZx(E[1]), 'u')):
if EF.degree() == n and EF.disc() <= B:
S.append(
[EF.disc(),
pari(EF.absolute_polynomial())])
S += enumerate_totallyreal_fields_prim(n, B, verbose=verbose)
S.sort(key=lambda x: (x[0], [QQ(cf) for cf in x[1].polrecip().Vec()]))
weed_fields(S)
# Output.
if verbose:
saveout = sys.stdout
if isinstance(verbose, str):
fsock = open(verbose, 'w')
sys.stdout = fsock
# Else, print to screen
print("=" * 80)
print("Polynomials tested: {}".format(counts[0]))
print("Polynomials with discriminant with large enough square"
" divisor: {}".format(counts[1]))
print("Irreducible polynomials: {}".format(counts[2]))
print("Polynomials with nfdisc <= B: {}".format(counts[3]))
for i in range(len(S)):
print(S[i])
if isinstance(verbose, str):
fsock.close()
sys.stdout = saveout
# Make sure to return elements that belong to Sage
if return_seqs:
return [[ZZ(_) for _ in counts],
[[ZZ(s[0]), [QQ(_) for _ in s[1].polrecip().Vec()]]
for s in S]]
elif return_pari_objects:
return S
else:
Px = PolynomialRing(QQ, 'x')
return [[ZZ(s[0]), Px([QQ(_) for _ in s[1].list()])] for s in S]
def enumerate_totallyreal_fields_rel(F,
m,
B,
a=[],
verbose=0,
return_seqs=False,
return_pari_objects=True):
r"""
This function enumerates (primitive) totally real field extensions of
degree `m>1` of the totally real field F with discriminant `d \leq B`;
optionally one can specify the first few coefficients, where the sequence ``a``
corresponds to a polynomial by
::
a[d]*x^n + ... + a[0]*x^(n-d)
if ``length(a) = d+1``, so in particular always ``a[d] = 1``.
.. note::
This is guaranteed to give all primitive such fields, and
seems in practice to give many imprimitive ones.
INPUT:
- ``F`` -- number field, the base field
- ``m`` -- integer, the degree
- ``B`` -- integer, the discriminant bound
- ``a`` -- list (default: []), the coefficient list to begin with
- ``verbose`` -- boolean or nonnegative integer or string (default: 0)
give a verbose description of the computations being performed. If
``verbose`` is set to ``2`` or more then it outputs some extra
information. If ``verbose`` is a string then it outputs to a file
specified by ``verbose``
- ``return_seqs`` -- (boolean, default False) If ``True``, then return
the polynomials as sequences (for easier exporting to a file). This
also returns a list of four numbers, as explained in the OUTPUT
section below.
- ``return_pari_objects`` -- (boolean, default: True) if
both ``return_seqs`` and ``return_pari_objects`` are ``False`` then
it returns the elements as Sage objects; otherwise it returns pari
objects.
OUTPUT:
- the list of fields with entries ``[d,fabs,f]``, where ``d`` is the
discriminant, ``fabs`` is an absolute defining polynomial, and ``f``
is a defining polynomial relative to ``F``, sorted by discriminant.
- if ``return_seqs`` is ``True``, then the first field of the list is
a list containing the count of four items as explained below
- the first entry gives the number of polynomials tested
- the second entry gives the number of polynomials with its
discriminant having a large enough square divisor
- the third entry is the number of irreducible polynomials
- the fourth entry is the number of irreducible polynomials with
discriminant at most ``B``
EXAMPLES::
sage: ZZx = ZZ['x']
sage: F.<t> = NumberField(x^2-2)
sage: enumerate_totallyreal_fields_rel(F, 1, 2000)
[[1, [-2, 0, 1], xF - 1]]
sage: enumerate_totallyreal_fields_rel(F, 2, 2000)
[[1600, x^4 - 6*x^2 + 4, xF^2 + xF - 1]]
sage: enumerate_totallyreal_fields_rel(F, 2, 2000, return_seqs=True)
[[9, 6, 5, 0], [[1600, [4, 0, -6, 0, 1], [-1, 1, 1]]]]
TESTS:
Each of the outputs must be elements of Sage if ``return_pari_objects``
is set to ``False``::
sage: type(enumerate_totallyreal_fields_rel(F, 2, 2000)[0][1])
<type 'cypari2.gen.Gen'>
sage: enumerate_totallyreal_fields_rel(F, 2, 2000, return_pari_objects=False)[0][0].parent()
Integer Ring
sage: enumerate_totallyreal_fields_rel(F, 2, 2000, return_pari_objects=False)[0][1].parent()
Univariate Polynomial Ring in x over Rational Field
sage: enumerate_totallyreal_fields_rel(F, 2, 2000, return_pari_objects=False)[0][2].parent()
Univariate Polynomial Ring in xF over Number Field in t with defining polynomial x^2 - 2
sage: enumerate_totallyreal_fields_rel(F, 2, 2000, return_seqs=True)[1][0][1][0].parent()
Rational Field
AUTHORS:
- John Voight (2007-11-01)
"""
if not isinstance(m, Integer):
try:
m = Integer(m)
except TypeError:
raise TypeError("cannot coerce m (= %s) to an integer" % m)
if (m < 1):
raise ValueError("m must be at least 1.")
n = F.degree() * m
# Initialize
S = {} # dictionary of the form {(d, fabs): f, ...}
dB_odlyzko = odlyzko_bound_totallyreal(n)
dB = math.ceil(40000 * dB_odlyzko**n)
counts = [0, 0, 0, 0]
# Trivial case
if m == 1:
g = pari(F.defining_polynomial()).polrecip().Vec()
if return_seqs:
return [[0, 0, 0, 0], [1, [-1, 1], g]]
elif return_pari_objects:
return [[1, g, pari('xF-1')]]
else:
Px = PolynomialRing(QQ, 'xF')
return [[ZZ(1), [QQ(_) for _ in g], Px.gen() - 1]]
if verbose:
saveout = sys.stdout
if isinstance(verbose, str):
fsock = open(verbose, 'w')
sys.stdout = fsock
# Else, print to screen
f_out = [0] * m + [1]
T = tr_data_rel(F, m, B, a)
if verbose == 2:
T.incr(f_out, verbose)
else:
T.incr(f_out)
Fx = PolynomialRing(F, 'xF')
nfF = pari(
str(F.defining_polynomial()).replace('x', str(F.primitive_element())))
parit = pari(str(F.primitive_element()))
while f_out[m] != 0:
counts[0] += 1
if verbose:
print("==>", f_out, end="")
f_str = ''
for i in range(len(f_out)):
f_str += '(' + str(f_out[i]) + ')*x^' + str(i)
if i < len(f_out) - 1:
f_str += '+'
nf = pari(f_str)
if nf.poldegree('t') == 0:
nf = nf.subst('x', 'x-t')
nf = nf.polresultant(nfF, parit)
d = nf.poldisc()
#counts[0] += 1
if d > 0 and nf.polsturm() == n:
da = int_has_small_square_divisor(Integer(d))
if d > dB or d <= B * da:
counts[1] += 1
if nf.polisirreducible():
counts[2] += 1
[zk, d] = nf.nfbasis_d()
if d <= B:
if verbose:
print("has discriminant", d, end="")
# Find a minimal lattice element
counts[3] += 1
ng = pari([nf, zk]).polredabs()
# Check if K is contained in the list.
if (d, ng) in S:
if verbose:
print("but is not new")
else:
if verbose:
print("and is new!")
S[(d, ng)] = Fx(f_out)
else:
if verbose:
print("has discriminant", abs(d), "> B")
else:
if verbose:
print("is not absolutely irreducible")
else:
if verbose:
print("has discriminant", abs(d),
"with no large enough square divisor")
else:
if verbose:
if d == 0:
print("is not squarefree")
else:
print("is not totally real")
if verbose == 2:
T.incr(f_out, verbose=verbose)
else:
T.incr(f_out)
# In the application of Smyth's theorem above, we exclude finitely
# many possibilities which we must now throw back in.
if m == 2:
if Fx([-1, 1, 1]).is_irreducible():
K = F.extension(Fx([-1, 1, 1]), 'tK')
Kabs = K.absolute_field('tKabs')
Kabs_pari = pari(Kabs.defining_polynomial())
d = K.absolute_discriminant()
if abs(d) <= B:
ng = Kabs_pari.polredabs()
S[(d, ng)] = Fx([-1, 1, 1])
elif F.degree() == 2:
for ff in [[1, -7, 13, -7, 1], [1, -8, 14, -7, 1]]:
f = Fx(ff).factor()[0][0]
K = F.extension(f, 'tK')
Kabs = K.absolute_field('tKabs')
Kabs_pari = pari(Kabs.defining_polynomial())
d = K.absolute_discriminant()
if abs(d) <= B:
ng = Kabs_pari.polredabs()
S[(d, ng)] = f
elif m == 3:
if Fx([-1, 6, -5, 1]).is_irreducible():
K = F.extension(Fx([-1, 6, -5, 1]), 'tK')
Kabs = K.absolute_field('tKabs')
Kabs_pari = pari(Kabs.defining_polynomial())
d = K.absolute_discriminant()
if abs(d) <= B:
ng = Kabs_pari.polredabs()
S[(d, ng)] = Fx([-1, 6, -5, 1])
# Convert S to a sorted list of triples [d, fabs, f], taking care
# not to use the comparison operators on PARI polynomials.
S = [[s[0], s[1], t] for s, t in S.items()]
S.sort(key=lambda x: (x[0], [QQ(cf) for cf in x[1].polrecip().Vec()]))
# Now check for isomorphic fields
weed_fields(S)
# Output.
if verbose:
print("=" * 80)
print("Polynomials tested: {}".format(counts[0]))
print("Polynomials with discriminant with large enough square"
" divisor: {}".format(counts[1]))
print("Irreducible polynomials: {}".format(counts[2]))
print("Polynomials with nfdisc <= B: {}".format(counts[3]))
for i in range(len(S)):
print(S[i])
if isinstance(verbose, str):
fsock.close()
sys.stdout = saveout
# Make sure to return elements that belong to Sage
if return_seqs:
return [[ZZ(x) for x in counts],
[[
s[0], [QQ(x) for x in s[1].polrecip().Vec()],
s[2].coefficients(sparse=False)
] for s in S]]
elif return_pari_objects:
return S
else:
Px = PolynomialRing(QQ, 'x')
return [[s[0], Px([QQ(_) for _ in s[1].list()]), s[2]] for s in S]
def enumerate_totallyreal_fields_all(n, B, verbose=0, return_seqs=False):
r"""
Enumerates *all* totally real fields of degree `n` with discriminant `\le B`,
primitive or otherwise.
EXAMPLES::
sage: enumerate_totallyreal_fields_all(4, 2000)
[[725, x^4 - x^3 - 3*x^2 + x + 1],
[1125, x^4 - x^3 - 4*x^2 + 4*x + 1],
[1600, x^4 - 6*x^2 + 4],
[1957, x^4 - 4*x^2 - x + 1],
[2000, x^4 - 5*x^2 + 5]]
In practice most of these will be found by :func:`~sage.rings.number_field.totallyreal.enumerate_totallyreal_fields_prim`, which is guaranteed to return all primitive fields but often returns many non-primitive ones as well. For instance, only one of the five fields in the example above is primitive, but :func:`~sage.rings.number_field.totallyreal.enumerate_totallyreal_fields_prim` finds four out of the five (the exception being `x^4 - 6x^2 + 4`).
"""
S = []
counts = [0, 0, 0]
if len(divisors(n)) > 4:
raise ValueError, "Only implemented for n = p*q with p,q prime"
for d in divisors(n):
if d > 1 and d < n:
Sds = enumerate_totallyreal_fields_prim(d, int(math.floor((1.0 * B) ** (1.0 * d / n))), verbose=verbose)
for i in range(len(Sds)):
if verbose:
print "=" * 80
print "Taking F =", Sds[i][1]
F = NumberField(ZZx(Sds[i][1]), "t")
T = enumerate_totallyreal_fields_rel(F, n / d, B, verbose=verbose, return_seqs=return_seqs)
if return_seqs:
for i in range(3):
counts[i] += T[0][i]
S += [[t[0], pari(t[1]).Polrev()] for t in T[1]]
else:
S += [[t[0], t[1]] for t in T]
j = i + 1
for E in enumerate_totallyreal_fields_prim(
n / d, int(math.floor((1.0 * B) ** (1.0 / d) / (1.0 * Sds[i][0]) ** (n * 1.0 / d ** 2)))
):
for EF in F.composite_fields(NumberField(ZZx(E[1]), "u")):
if EF.degree() == n and EF.disc() <= B:
S.append([EF.disc(), pari(EF.absolute_polynomial())])
S += enumerate_totallyreal_fields_prim(n, B, verbose=verbose)
S.sort()
weed_fields(S)
# Output.
if verbose:
saveout = sys.stdout
if type(verbose) == str:
fsock = open(verbose, "w")
sys.stdout = fsock
# Else, print to screen
print "=" * 80
print "Polynomials tested:", counts[0]
print "Irreducible polynomials:", counts[1]
print "Polynomials with nfdisc <= B:", counts[2]
for i in range(len(S)):
print S[i]
if type(verbose) == str:
fsock.close()
sys.stdout = saveout
if return_seqs:
return [counts, [[s[0], s[1].reverse().Vec()] for s in S]]
else:
return S
def enumerate_totallyreal_fields_rel(F, m, B, a=[], verbose=0, return_seqs=False):
r"""
This function enumerates (primitive) totally real field extensions of
degree `m>1` of the totally real field F with discriminant `d \leq B`;
optionally one can specify the first few coefficients, where the sequence ``a``
corresponds to a polynomial by
::
a[d]*x^n + ... + a[0]*x^(n-d)
if ``length(a) = d+1``, so in particular always ``a[d] = 1``.
If verbose == 1 (or 2), then print to the screen (really) verbosely; if
verbose is a string, then print verbosely to the file specified by verbose.
If return_seqs, then return the polynomials as sequences (for easier
exporting to a file).
NOTE:
This is guaranteed to give all primitive such fields, and
seems in practice to give many imprimitive ones.
INPUT:
- ``F`` -- number field, the base field
- ``m`` -- integer, the degree
- ``B`` -- integer, the discriminant bound
- ``a`` -- list (default: []), the coefficient list to begin with
- ``verbose`` -- boolean or string (default: 0)
- ``return_seqs`` -- boolean (default: False)
OUTPUT:
the list of fields with entries [d,fabs,f], where
d is the discriminant, fabs is an absolute defining polynomial,
and f is a defining polynomial relative to F,
sorted by discriminant.
EXAMPLES::
sage: ZZx = ZZ['x']
sage: F.<t> = NumberField(x^2-2)
sage: enumerate_totallyreal_fields_rel(F, 2, 2000)
[[1600, x^4 - 6*x^2 + 4, xF^2 + xF - 1]]
AUTHORS:
- John Voight (2007-11-01)
"""
if not isinstance(m, Integer):
try:
m = Integer(m)
except TypeError:
raise TypeError, "cannot coerce m (= %s) to an integer" % m
if m < 1:
raise ValueError, "m must be at least 1."
n = F.degree() * m
# Initialize
T = tr_data_rel(F, m, B, a)
S = []
Srel = []
dB_odlyzko = odlyzko_bound_totallyreal(n)
dB = math.ceil(40000 * dB_odlyzko ** n)
counts = [0, 0, 0, 0]
# Trivial case
if m == 1:
g = pari(F.defining_polynomial()).reverse().Vec()
if return_seqs:
return [[0, 0, 0, 0], [1, g, [-1, 1]]]
else:
return [[1, pari("x-1"), g]]
if verbose:
saveout = sys.stdout
if type(verbose) == str:
fsock = open(verbose, "w")
sys.stdout = fsock
# Else, print to screen
f_out = [0] * m + [1]
if verbose == 2:
T.incr(f_out, verbose)
else:
T.incr(f_out)
Fx = PolynomialRing(F, "xF")
nfF = pari(str(F.defining_polynomial()).replace("x", str(F.primitive_element())))
parit = pari(str(F.primitive_element()))
while f_out[m] <> 0:
counts[0] += 1
if verbose:
print "==>", f_out,
f_str = ""
for i in range(len(f_out)):
f_str += "(" + str(f_out[i]) + ")*x^" + str(i)
if i < len(f_out) - 1:
f_str += "+"
nf = pari(f_str)
if nf.poldegree("t") == 0:
nf = nf.subst("x", "x-t")
nf = nf.polresultant(nfF, parit)
d = nf.poldisc()
counts[0] += 1
if d > 0 and nf.polsturm_full() == n:
da = int_has_small_square_divisor(Integer(d))
if d > dB or d <= B * da:
counts[1] += 1
if nf.polisirreducible():
counts[2] += 1
[zk, d] = nf.nfbasis_d()
if d <= B:
if verbose:
print "has discriminant", d,
# Find a minimal lattice element
counts[3] += 1
ng = pari([nf, zk]).polredabs()
# Check if K is contained in the list.
found = False
ind = bisect.bisect_left(S, [d, ng])
while ind < len(S) and S[ind][0] == d:
if S[ind][1] == ng:
if verbose:
print "but is not new"
found = True
break
ind += 1
if not found:
if verbose:
print "and is new!"
S.insert(ind, [d, ng])
Srel.insert(ind, Fx(f_out))
else:
if verbose:
print "has discriminant", abs(d), "> B"
else:
if verbose:
print "is not absolutely irreducible"
else:
if verbose:
print "has discriminant", abs(d), "with no large enough square divisor"
else:
if verbose:
if d == 0:
print "is not squarefree"
else:
print "is not totally real"
if verbose == 2:
T.incr(f_out, verbose=verbose)
else:
T.incr(f_out)
# In the application of Smyth's theorem above, we exclude finitely
# many possibilities which we must now throw back in.
if m == 2:
if Fx([-1, 1, 1]).is_irreducible():
K = F.extension(Fx([-1, 1, 1]), "tK")
Kabs = K.absolute_field("tKabs")
Kabs_pari = pari(Kabs.defining_polynomial())
d = K.absolute_discriminant()
if abs(d) <= B:
ng = Kabs_pari.polredabs()
ind = bisect.bisect_left(S, [d, ng])
S.insert(ind, [d, ng])
Srel.insert(ind, Fx([-1, 1, 1]))
elif F.degree() == 2:
for ff in [[1, -7, 13, -7, 1], [1, -8, 14, -7, 1]]:
f = Fx(ff).factor()[0][0]
K = F.extension(f, "tK")
Kabs = K.absolute_field("tKabs")
Kabs_pari = pari(Kabs.defining_polynomial())
d = K.absolute_discriminant()
if abs(d) <= B:
ng = Kabs_pari.polredabs()
ind = bisect.bisect_left(S, [d, ng])
S.insert(ind, [d, ng])
Srel.insert(ind, f)
elif m == 3:
if Fx([-1, 6, -5, 1]).is_irreducible():
K = F.extension(Fx([-1, 6, -5, 1]), "tK")
Kabs = K.absolute_field("tKabs")
Kabs_pari = pari(Kabs.defining_polynomial())
d = K.absolute_discriminant()
if abs(d) <= B:
ng = Kabs_pari.polredabs()
ind = bisect.bisect_left(S, [d, ng])
S.insert(ind, [d, ng])
Srel.insert(ind, Fx([-1, 6, -5, 1]))
# Now check for isomorphic fields
S = [[S[i][0], S[i][1], Srel[i]] for i in range(len(S))]
weed_fields(S)
# Output.
if verbose:
print "=" * 80
print "Polynomials tested:", counts[0]
print "Irreducible polynomials:", counts[1]
print "Polynomials with nfdisc <= B:", counts[2]
for i in range(len(S)):
print S[i]
if type(verbose) == str:
fsock.close()
sys.stdout = saveout
if return_seqs:
return [counts, [[s[0], s[1].reverse().Vec(), s[2].coeffs()] for s in S]]
else:
return S
