def assertIsReduced(t, m_path, x_name, eps_name):
M = fuchsia.import_matrix_from_file(m_path)
x = SR.var(x_name)
eps = SR.var(eps_name)
pranks = fuchsia.singularities(M, x).values()
t.assertEqual(pranks, [0]*len(pranks))
t.assertTrue(eps not in (M/eps).simplify_rational().variables())
def test_normalize_3(t):
# Test with non-zero normalized eigenvalues
x = SR.var("x")
e = SR.var("epsilon")
M = matrix([[(1 - e) / x, 0], [0, (1 + e) / 3 / x]])
with t.assertRaises(FuchsiaError):
N, T = normalize(M, x, e)
def test_factorize_1(t):
x = SR.var("x")
e = SR.var("epsilon")
M = matrix([[1 / x, 0, 0], [0, 2 / x, 0], [0, 0, 3 / x]]) * e
M = transform(M, x, matrix([[1, 1, 0], [0, 1, 0], [1 + 2 * e, 0, e]]))
F, T = factorize(M, x, e)
F = F.simplify_rational()
for f in F.list():
t.assertEqual(limit_fixed(f, e, 0), 0)
def test_is_normalized_1(t):
x = SR.var("x")
e = SR.var("epsilon")
t.assertFalse(is_normalized(matrix([[1 / x / 2]]), x, e))
t.assertFalse(is_normalized(matrix([[-1 / x / 2]]), x, e))
t.assertTrue(is_normalized(matrix([[1 / x / 3]]), x, e))
t.assertFalse(is_normalized(matrix([[x]]), x, e))
t.assertFalse(is_normalized(matrix([[1 / x**2]]), x, e))
t.assertTrue (is_normalized( \
matrix([[(e+SR(1)/3)/x-SR(1)/2/(x-1)]]), x, e))
def test_normalize_1(t):
# Test with apparent singularities at 0 and oo, but not at 1.
x = SR.var("x")
M = matrix([[1 / x, 5 / (x - 1), 0, 6 / (x - 1)], [0, 2 / x, 0, 0],
[0, 0, 3 / x, 7 / (x - 1)], [6 / (x - 1), 0, 0, 1 / x]])
N, T = normalize(M, x, SR.var("epsilon"))
N = N.simplify_rational()
t.assertEqual(N, transform(M, x, T).simplify_rational())
for point, prank in singularities(N, x).iteritems():
R = matrix_c0(N, x, point, prank)
evlist = R.eigenvalues()
t.assertEqual(evlist, [0] * len(evlist))
def place_list(self, mx, Mx, frac=1, mark_origin=True):
"""
return a tuple of
1. values that are all the integer multiple of
<frac>*self.numerical_value
between mx and Mx
2. the multiple of the basis unit.
Give <frac> as literal real. Recall that python evaluates 1/2 to 0. If you pass 0.5, it will be converted back to 1/2 for a nice display.
"""
try:
# If the user enters "0.5", it is converted to 1/2
frac = Rational(frac)
except TypeError:
pass
if frac == 0:
raise ValueError(
"frac is zero in AxesUnit.place_list(). Maybe you ignore that python evaluates 1/2 to 0 ? (writes literal 0.5 instead) \n Or are you trying to push me in an infinite loop ?"
)
l = []
k = SR.var("TheTag")
for x in MultipleBetween(frac * self.numerical_value, mx, Mx,
mark_origin):
if self.latex_symbol == "":
l.append((x, "$" + latex(x) + "$"))
else:
pos = (x / self.numerical_value) * k
# This risks to be Sage-version dependent.
text = "$" + latex(pos).replace("TheTag",
self.latex_symbol) + "$"
l.append((x, text))
return l
def __init__(self,dof,name,shortname=None):
self.dof = int(dof)
self.name = name
if shortname == None :
self.shortname = name.replace(' ','_').replace('.','_')
else :
self.shortname = shortname.replace(' ','_').replace('.','_')
for i in range(1 ,1 +dof):
var('q'+str(i),domain=RR)
(alpha , a , d , theta) = SR.var('alpha,a,d,theta',domain=RR)
default_params_dh = [ alpha , a , d , theta ]
default_T_dh = matrix( \
[[ cos(theta), -sin(theta)*cos(alpha), sin(theta)*sin(alpha), a*cos(theta) ], \
[ sin(theta), cos(theta)*cos(alpha), -cos(theta)*sin(alpha), a*sin(theta) ], \
[ 0.0, sin(alpha), cos(alpha), d ]
,[ 0.0, 0.0, 0.0, 1.0] ] )
#g_a = SR.var('g_a',domain=RR)
g_a = 9.81
self.grav = matrix([[0.0],[0.0],[-g_a]])
self.params_dh = default_params_dh
self.T_dh = default_T_dh
self.links_dh = []
self.motors = []
self.Imzi = []
self._gensymbols()
def test_normalize_4(t):
# Test with non-zero normalized eigenvalues
x, e = SR.var("x eps")
M = matrix([[1 / x / 2, 0], [0, 0]])
with t.assertRaises(FuchsiaError):
N, T = normalize(M, x, e)
def assertReductionWorks(t, filename):
M = import_matrix_from_file(filename)
x, eps = SR.var("x eps")
t.assertIn(x, M.variables())
M_pranks = singularities(M, x).values()
t.assertNotEqual(M_pranks, [0] * len(M_pranks))
#1 Fuchsify
m, t1 = simplify_by_factorization(M, x)
Mf, t2 = fuchsify(m, x)
Tf = t1 * t2
t.assertTrue((Mf - transform(M, x, Tf)).simplify_rational().is_zero())
Mf_pranks = singularities(Mf, x).values()
t.assertEqual(Mf_pranks, [0] * len(Mf_pranks))
#2 Normalize
t.assertFalse(is_normalized(Mf, x, eps))
m, t1 = simplify_by_factorization(Mf, x)
Mn, t2 = normalize(m, x, eps)
Tn = t1 * t2
t.assertTrue((Mn - transform(Mf, x, Tn)).simplify_rational().is_zero())
t.assertTrue(is_normalized(Mn, x, eps))
#3 Factorize
t.assertIn(eps, Mn.variables())
m, t1 = simplify_by_factorization(Mn, x)
Mc, t2 = factorize(m, x, eps, seed=3)
Tc = t1 * t2
t.assertTrue((Mc - transform(Mn, x, Tc)).simplify_rational().is_zero())
t.assertNotIn(eps, (Mc / eps).simplify_rational().variables())
def test_reduce_at_one_point_1(t):
x = SR.var("x")
M0 = matrix([[1 / x, 4, 0, 5], [0, 2 / x, 0, 0], [0, 0, 3 / x, 6],
[0, 0, 0, 4 / x]])
u = matrix([[0, Rational((3, 5)),
Rational((4, 5)), 0],
[Rational((5, 13)), 0, 0,
Rational((12, 13))]])
M1 = transform(M0, x, balance(u.transpose() * u, 0, 1, x))
M1 = M1.simplify_rational()
u = matrix([[8, 0, 15, 0]]) / 17
M2 = transform(M1, x, balance(u.transpose() * u, 0, 2, x))
M2 = M2.simplify_rational()
M2_sing = singularities(M2, x)
t.assertIn(0, M2_sing)
t.assertEqual(M2_sing[0], 2)
M3, T23 = reduce_at_one_point(M2, x, 0, 2)
M3 = M3.simplify_rational()
t.assertEqual(M3, transform(M2, x, T23).simplify_rational())
M3_sing = singularities(M3, x)
t.assertIn(0, M3_sing)
t.assertEqual(M3_sing[0], 1)
M4, T34 = reduce_at_one_point(M3, x, 0, 1)
M4 = M4.simplify_rational()
t.assertEqual(M4, transform(M3, x, T34).simplify_rational())
M4_sing = singularities(M4, x)
t.assertIn(0, M4_sing)
t.assertEqual(M4_sing[0], 0)
def test_import_export_mathematica(t):
a, b = SR.var("v1 v2")
M = matrix([[1, a, b], [a + b, Rational((2, 3)), a / b]])
fout = StringIO()
export_matrix_mathematica(fout, M)
MM = import_matrix_mathematica(StringIO(fout.getvalue()))
t.assertEqual(M, MM)
def taylor_processor_factored(new_ring, Phi, scalar, alpha, I, omega):
k = alpha.nrows() - 1
tau = SR.var('tau')
y = [SR('y%d' % i) for i in range(k + 1)]
R = PolynomialRing(QQ, len(y), y)
beta = [a * Phi for a in alpha]
ell = len(I)
def f(i):
if i == 0:
return QQ(scalar) * y[0] * exp(tau * omega[0])
elif i in I:
return tau / (1 - exp(tau * omega[i]))
else:
return 1 / (1 - y[i] * exp(tau * omega[i]))
H = [
f(i).series(tau, ell + 1).truncate().collect(tau) for i in range(k + 1)
]
for i in range(k + 1):
H[i] = [H[i].coefficient(tau, j) for j in range(ell + 1)]
r = []
# Get coefficient of tau^ell in prod(H)
for w in NonnegativeCompositions(ell, k + 1):
r = prod(
CyclotomicRationalFunction.from_split_expression(H[i][
w[i]], y, R).monomial_substitution(new_ring, beta)
for i in range(k + 1))
yield r
def padically_evaluate_regular(self, datum):
T = datum.toric_datum
if not T.is_regular():
raise ValueError('Can only processed regular toric data')
M = Set(range(T.length()))
q = SR.var('q')
alpha = {}
for I in Subsets(M):
F = [T.initials[i] for i in I]
V = SubvarietyOfTorus(F, torus_dim=T.ambient_dim)
alpha[I] = V.count()
def cat(u, v):
return vector(list(u) + list(v))
for I in Subsets(M):
cnt = sum((-1)**len(J) * alpha[I + J] for J in Subsets(M - I))
if not cnt:
continue
P = DirectProductOfPolyhedra(T.polyhedron,
StrictlyPositiveOrthant(len(I)))
it = iter(identity_matrix(ZZ, len(I)).rows())
ieqs = []
for i in M:
ieqs.append(
cat(
vector(ZZ, (0, )),
cat(
vector(ZZ, T.initials[i].exponents()[0]) -
vector(ZZ, T.lhs[i].exponents()[0]),
next(it) if i in I else zero_vector(ZZ, len(I)))))
if not ieqs:
ieqs = [vector(ZZ, (T.ambient_dim + len(I) + 1) * [0])]
Q = Polyhedron(ieqs=ieqs,
base_ring=QQ,
ambient_dim=T.ambient_dim + len(I))
foo, ring = symbolic_to_ratfun(
cnt * (q - 1)**len(I) / q**(T.ambient_dim),
[var('t'), var('q')])
corr_cnt = CyclotomicRationalFunction.from_laurent_polynomial(
foo, ring)
Phi = matrix([
cat(T.integrand[0], zero_vector(ZZ, len(I))),
cat(T.integrand[1], vector(ZZ,
len(I) * [-1]))
]).transpose()
sm = RationalSet([P.intersection(Q)]).generating_function()
for z in sm.monomial_substitution(QQ['t', 'q'], Phi):
yield corr_cnt * z
def test_transform_2(t):
# transform(transform(M, x, I), x, I^-1) == M
x = SR.var("x")
M = randpolym(x, 2)
T = randpolym(x, 2)
invT = T.inverse()
M1 = transform(M, x, T)
M2 = transform(M1, x, invT)
t.assertEqual(M2.simplify_rational(), M)
def test_fuchsify_by_blocks_03(test):
x, eps = SR.var("x eps")
m = matrix([
[eps / x, 0, 0],
[1 / x**2, 2 * eps / x, 0],
[1 / x**2, 2 / x**2, 3 * eps / x],
])
b = [(0, 1), (1, 1), (2, 1)]
test.assert_fuchsify_by_blocks_works(m, b, x, eps)
def test_fuchsify_by_blocks_05(test):
x, eps = SR.var("x eps")
m = matrix([
[eps / x, 0, 0],
[0, 2 * eps / x, -eps / x],
[x**2, 0, 3 * eps / x],
])
b = [(0, 1), (1, 2)]
test.assert_fuchsify_by_blocks_works(m, b, x, eps)
def test_normalize_5(t):
# An unnormalizable example by A. A. Bolibrukh
x, e = SR.var("x eps")
b = import_matrix_from_file("test/data/bolibrukh.mtx")
f, ft = fuchsify(b, x)
f_pranks = singularities(f, x).values()
t.assertEqual(f_pranks, [0] * len(f_pranks))
with t.assertRaises(FuchsiaError):
n, nt = normalize(f, x, e)
def get_background_graphic(self, **bdry_options):
r"""
Return a graphic object that makes the model easier to visualize.
For the hyperboloid model, the background object is the hyperboloid
itself.
EXAMPLES::
sage: H = HyperbolicPlane().HM().get_background_graphic()
"""
from sage.plot.plot3d.all import plot3d
from sage.all import SR
hyperboloid_opacity = bdry_options.get('hyperboloid_opacity', .1)
z_height = bdry_options.get('z_height', 7.0)
x_max = sqrt((z_height ** 2 - 1) / 2.0)
x = SR.var('x')
y = SR.var('y')
return plot3d((1 + x ** 2 + y ** 2).sqrt(),
(x, -x_max, x_max), (y,-x_max, x_max),
opacity=hyperboloid_opacity, **bdry_options)
def get_background_graphic(self, **bdry_options):
r"""
Return a graphic object that makes the model easier to visualize.
For the hyperboloid model, the background object is the hyperboloid
itself.
EXAMPLES::
sage: H = HyperbolicPlane().HM().get_background_graphic()
"""
from sage.plot.plot3d.all import plot3d
from sage.all import SR
hyperboloid_opacity = bdry_options.get('hyperboloid_opacity', .1)
z_height = bdry_options.get('z_height', 7.0)
x_max = sqrt((z_height ** 2 - 1) / 2.0)
x = SR.var('x')
y = SR.var('y')
return plot3d((1 + x ** 2 + y ** 2).sqrt(),
(x, -x_max, x_max), (y,-x_max, x_max),
opacity=hyperboloid_opacity, **bdry_options)
def test_balance_2(t):
# balance(P, x1, oo, x)*balance(P, oo, x1, x) == I
x = SR.var("x")
P = matrix([[1, 1], [0, 0]])
x1 = randint(-10, 10)
b1 = balance(P, x1, oo, x)
b2 = balance(P, oo, x1, x)
t.assertEqual((b1 * b2).simplify_rational(),
identity_matrix(P.nrows()))
t.assertEqual((b2 * b1).simplify_rational(),
identity_matrix(P.nrows()))
def test_fuchsify_by_blocks_06(test):
x, eps = SR.var("x eps")
m = matrix(
[[eps / (x - 1), 0, 0],
[(x**3 + x**2 - 3 * x - 3) / (2 * x**3 - 2 * x**2 - x + 1),
2 * eps / (x - 3), 0],
[
-(x**3 + 3 * x**2 - 3 * x + 2) / (2 * x**3 + x**2 - 2 * x) +
2 * (x**3 - x**2 - x + 1) / (2 * x + 1),
-3 * (x**3 + x - 1) / (3 * x**3 + x - 2), 4 * eps / (x - 5)
]])
b = [(0, 1), (1, 1), (2, 1)]
test.assert_fuchsify_by_blocks_works(m, b, x, eps)
def assertReductionWorks(test, filename, fuchsian=False):
m = import_matrix_from_file(filename)
x, eps = SR.var("x eps")
test.assertIn(x, m.variables())
if not fuchsian:
m_pranks = singularities(m, x).values()
test.assertNotEqual(m_pranks, [0]*len(m_pranks))
mt, t = epsilon_form(m, x, eps)
test.assertTrue((mt-transform(m, x, t)).simplify_rational().is_zero())
test.assertTrue(is_fuchsian(mt, x))
test.assertTrue(is_normalized(mt, x, eps))
test.assertNotIn(eps, (mt/eps).simplify_rational().variables())
def test_block_triangular_form_3(t):
m = matrix([
[1, 0, 1, 0],
[0, 1, 0, 1],
[1, 0, 1, 0],
[0, 0, 0, 1]
])
mt, tt, b = block_triangular_form(m)
x = SR.var("dummy")
t.assertEqual(mt, transform(m, x, tt))
t.assertEqual(matrix([
[1, 1, 0, 0],
[1, 1, 0, 0],
[0, 0, 1, 0],
[0, 0, 1, 1]
]), mt)
def assertNormalizeBlocksWorks(test, filename):
x, eps = SR.var("x eps")
m = import_matrix_from_file(filename)
test.assertIn(x, m.variables())
test.assertIn(eps, m.variables())
test.assertFalse(is_normalized(m, x, eps))
m_pranks = singularities(m, x).values()
test.assertNotEqual(m_pranks, [0] * len(m_pranks))
m, t, b = block_triangular_form(m)
mt, tt = reduce_diagonal_blocks(m, x, eps, b=b)
t = t * tt
test.assertTrue(
(mt - transform(m, x, t)).simplify_rational().is_zero())
test.assertTrue(are_diagonal_blocks_reduced(mt, b, x, eps))
def test_simplify_by_jordanification(t):
x = SR.var("x")
M = matrix(
[[4 / (x + 1), -1 / (6 * x * (x + 1)), -1 / (3 * x * (x + 1))],
[
6 * (13 * x + 6) / (x * (x + 1)),
-5 * (x + 3) / (3 * x * (x + 1)),
2 * (x - 6) / (3 * x * (x + 1))
],
[
-63 * (x - 1) / (x * (x + 1)),
(5 * x - 9) / (6 * x * (x + 1)), -(x - 18) / (3 * x * (x + 1))
]]).simplify_rational()
MM, T = simplify_by_jordanification(M, x)
MM = MM.simplify_rational()
t.assertEqual(MM, transform(M, x, T).simplify_rational())
t.assertLess(matrix_complexity(MM), matrix_complexity(M))
def test_fuchsify_2(t):
x = SR.var("x")
M = matrix([[0, 1 / x / (x - 1), 0, 0], [0, 0, 0, 0], [0, 0, 0, 0],
[0, 0, 0, 0]])
u = matrix([[6, 3, 2, 0]]) / 7
P = u.transpose() * u
M = balance_transform(M, P, 1, 0, x).simplify_rational()
M = balance_transform(M, P, 1, 0, x).simplify_rational()
M = balance_transform(M, P, 1, 0, x).simplify_rational()
M = balance_transform(M, P, 1, 0, x).simplify_rational()
M = balance_transform(M, P, 1, 0, x).simplify_rational()
MM, T = fuchsify(M, x)
MM = MM.simplify_rational()
t.assertEqual(MM, transform(M, x, T).simplify_rational())
pranks = singularities(MM, x).values()
t.assertEqual(pranks, [0] * len(pranks))
def test_balance_transform_1(t):
x = SR.var("x")
M = randpolym(x, 2)
P = matrix([[1, 1], [0, 0]])
x1 = randint(-10, 10)
x2 = randint(20, 30)
b1 = balance(P, x1, x2, x)
M1 = balance_transform(M, P, x1, x2, x)
M2 = transform(M, x, balance(P, x1, x2, x))
t.assertEqual(M1.simplify_rational(), M2.simplify_rational())
M1 = balance_transform(M, P, x1, oo, x)
M2 = transform(M, x, balance(P, x1, oo, x))
t.assertEqual(M1.simplify_rational(), M2.simplify_rational())
M1 = balance_transform(M, P, oo, x2, x)
M2 = transform(M, x, balance(P, oo, x2, x))
t.assertEqual(M1.simplify_rational(), M2.simplify_rational())
def test_fuchsify_1(t):
x = SR.var("x")
M = matrix([[1 / x, 5, 0, 6], [0, 2 / x, 0, 0], [0, 0, 3 / x, 7],
[0, 0, 0, 4 / x]])
u = matrix([[0, Rational((3, 5)),
Rational((4, 5)), 0],
[Rational((5, 13)), 0, 0,
Rational((12, 13))]])
M = transform(M, x, balance(u.transpose() * u, 0, 1, x))
M = M.simplify_rational()
u = matrix([[8, 0, 15, 0]]) / 17
M = transform(M, x, balance(u.transpose() * u, 0, 2, x))
M = M.simplify_rational()
Mx, T = fuchsify(M, x)
Mx = Mx.simplify_rational()
t.assertEqual(Mx, transform(M, x, T).simplify_rational())
pranks = singularities(Mx, x).values()
t.assertEqual(pranks, [0] * len(pranks))
def taylor_processor_naive(new_ring, Phi, scalar, alpha, I, omega):
k = alpha.nrows() - 1
tau = SR.var('tau')
y = [SR('y%d' % i) for i in range(k + 1)]
R = PolynomialRing(QQ, len(y), y)
beta = [a * Phi for a in alpha]
def f(i):
if i == 0:
return QQ(scalar) * y[0] * exp(tau * omega[0])
elif i in I:
return 1 / (1 - exp(tau * omega[i]))
else:
return 1 / (1 - y[i] * exp(tau * omega[i]))
h = prod(f(i) for i in range(k + 1))
# Get constant term of h as a Laurent series in tau.
g = h.series(tau, 1).truncate().collect(tau).coefficient(tau, 0)
g = g.factor() if g else g
yield CyclotomicRationalFunction.from_split_expression(
g, y, R).monomial_substitution(new_ring, beta)
def test_block_triangular_form_4(t):
M = matrix([
[1, 2, 3, 0, 0, 0],
[4, 5, 6, 0, 0, 0],
[7, 8, 9, 0, 0, 0],
[2, 0, 0, 1, 2, 0],
[0, 2, 0, 3, 4, 0],
[0, 0, 2, 0, 0, 1]
])
x = SR.var("dummy")
T = matrix.identity(6)[random.sample(xrange(6), 6),:]
M = transform(M, x, T)
MM, T, B = block_triangular_form(M)
t.assertEqual(MM, transform(M, x, T))
t.assertEqual(sorted(s for o, s in B), [1, 2, 3])
for o, s in B:
for i in xrange(s):
for j in xrange(s):
MM[o + i, o + j] = 0
for i in xrange(6):
for j in xrange(i):
MM[i, j] = 0
t.assertEqual(MM, matrix(6))
def test_fuchsify_by_blocks_07(test):
x, eps = SR.var("x ep")
m = matrix([[0, 0], [1 / (x**2 - x + 1)**2, 0]])
b = [(0, 1), (1, 1)]
test.assert_fuchsify_by_blocks_works(m, b, x, eps)
def elliptic_cm_form(E, n, prec, aplist_only=False, anlist_only=False):
"""
Return q-expansion of the CM modular form associated to the n-th
power of the Grossencharacter associated to the elliptic curve E.
INPUT:
- E -- CM elliptic curve
- n -- positive integer
- prec -- positive integer
- aplist_only -- return list only of ap for p prime
- anlist_only -- return list only of an
OUTPUT:
- power series with integer coefficients
EXAMPLES::
sage: from psage.modform.rational.special import elliptic_cm_form
sage: f = CuspForms(121,4).newforms(names='a')[0]; f
q + 8*q^3 - 8*q^4 + 18*q^5 + O(q^6)
sage: E = EllipticCurve('121b')
sage: elliptic_cm_form(E, 3, 7)
q + 8*q^3 - 8*q^4 + 18*q^5 + O(q^7)
sage: g = elliptic_cm_form(E, 3, 100)
sage: g == f.q_expansion(100)
True
"""
if not E.has_cm():
raise ValueError, "E must have CM"
n = ZZ(n)
if n <= 0:
raise ValueError, "n must be positive"
prec = ZZ(prec)
if prec <= 0:
return []
elif prec <= 1:
return [ZZ(0)]
elif prec <= 2:
return [ZZ(0), ZZ(1)]
# Derive formula for sum of n-th powers of roots
a,p,T = SR.var('a,p,T')
roots = (T**2 - a*T + p).roots(multiplicities=False)
s = sum(alpha**n for alpha in roots).simplify_full()
# Create fast callable expression from formula
g = fast_callable(s.polynomial(ZZ))
# Compute aplist for the curve
v = E.aplist(prec)
# Use aplist to compute ap values for the CM form attached to n-th
# power of Grossencharacter.
P = prime_range(prec)
if aplist_only:
# case when we only want the a_p (maybe for computing an
# L-series via Euler product)
return [g(ap,p) for ap,p in zip(v,P)]
# Default cause where we want all a_n
anlist = [ZZ(0),ZZ(1)] + [None]*(prec-2)
for ap,p in zip(v,P):
anlist[p] = g(ap,p)
# Fill in the prime power a_{p^r} for r >= 2.
N = E.conductor()
for p in P:
prm2 = 1
prm1 = p
pr = p*p
pn = p**n
e = 1 if N%p else 0
while pr < prec:
anlist[pr] = anlist[prm1] * anlist[p]
if e:
anlist[pr] -= pn * anlist[prm2]
prm2 = prm1
prm1 = pr
pr *= p
# fill in a_n with n divisible by at least 2 primes
extend_multiplicatively_generic(anlist)
if anlist_only:
return anlist
f = Integer_list_to_polynomial(anlist, 'q')
return ZZ[['q']](f, prec=prec)
