Coverage for local/lib/python2.7/site-packages/sage/rings/polynomial/polynomial_number

Hot-keys on this page

r m x p toggle line displays

j k next/prev highlighted chunk

0 (zero) top of page

1 (one) first highlighted chunk

r"""

Univariate polynomials over number fields.

AUTHOR:

- Luis Felipe Tabera Alonso (2014-02): initial version.

EXAMPLES:

Define a polynomial over an absolute number field and perform basic

operations with them::

sage: N.<a> = NumberField(x^2-2)

sage: K.<x> = N[]

sage: f = x - a

sage: g = x^3 - 2*a + 1

sage: f*(x + a)

x^2 - 2

sage: f + g

x^3 + x - 3*a + 1

sage: g // f

x^2 + a*x + 2

sage: g % f

sage: factor(x^3 - 2*a*x^2 - 2*x + 4*a)

(x - 2*a) * (x - a) * (x + a)

sage: gcd(f, x - a)

x - a

Polynomials are aware of embeddings of the underlying field::

sage: x = var('x')

sage: Q7 = Qp(7)

sage: r1 = Q7(3 + 7 + 2*7^2 + 6*7^3 + 7^4 + 2*7^5 + 7^6 + 2*7^7 + 4*7^8 +\

6*7^9 + 6*7^10 + 2*7^11 + 7^12 + 7^13 + 2*7^15 + 7^16 + 7^17 +\

4*7^18 + 6*7^19)

sage: N.<b> = NumberField(x^2-2, embedding = r1)

sage: K.<t> = N[]

sage: f = t^3-2*t+1

sage: f(r1)

1 + O(7^20)

We can also construct polynomials over relative number fields::

sage: N.<i, s2> = QQ[I, sqrt(2)]

sage: K.<x> = N[]

sage: f = x - s2

sage: g = x^3 - 2*i*x^2 + s2*x

sage: f*(x + s2)

x^2 - 2

sage: f + g

x^3 - 2*I*x^2 + (sqrt2 + 1)*x - sqrt2

sage: g // f

x^2 + (-2*I + sqrt2)*x - 2*sqrt2*I + sqrt2 + 2

sage: g % f

-4*I + 2*sqrt2 + 2

sage: factor(i*x^4 - 2*i*x^2 + 9*i)

(I) * (x - I + sqrt2) * (x + I - sqrt2) * (x - I - sqrt2) * (x + I + sqrt2)

sage: gcd(f, x-i)

"""

#*****************************************************************************

# Distributed under the terms of the GNU General Public License (GPL)

# as published by the Free Software Foundation; either version 2 of

# the License, or (at your option) any later version.

# http://www.gnu.org/licenses/

#*****************************************************************************

from __future__ import absolute_import

from .polynomial_element_generic import Polynomial_generic_dense_field

from sage.rings.rational_field import QQ

from sage.structure.element import coerce_binop

from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing

class Polynomial_absolute_number_field_dense(Polynomial_generic_dense_field):

"""

Class of dense univariate polynomials over an absolute number field.

"""

def __init__(self, parent, x=None, check=True, is_gen=False, construct=False):

"""

Create a new polynomial in the polynomial ring ``parent``.

INPUT:

- ``parent`` -- the polynomial ring in which to construct the

element.

- ``x`` -- (default: None) an object representing the

polynomial, e.g. a list of coefficients. See

:meth:`sage.rings.polynomial.polynomial_element_generic.Polynomial_generic_dense_field.__init__`

for more details.

- ``check`` -- boolean (default: True) if True, make sure that

the coefficients of the polynomial are in the base ring.

- ``is_gen`` -- boolean (default: False) if True, `x` is the

distinguished generator of the polynomial ring.

- ``construct`` -- (default: False) boolean, unused.

EXAMPLES::

sage: P.<x> = QQ[I][]

sage: f = P.random_element()

sage: from sage.rings.polynomial.polynomial_number_field import Polynomial_absolute_number_field_dense

sage: isinstance(f, Polynomial_absolute_number_field_dense)

True

sage: a = P(x)

sage: a.is_gen()

True

"""

Polynomial_generic_dense_field.__init__(self, parent, x, check, is_gen, construct)

@coerce_binop

def gcd(self, other):

"""

Compute the monic gcd of two univariate polynomials using PARI.

INPUT:

- ``other`` -- a polynomial with the same parent as ``self``.

OUTPUT:

- The monic gcd of ``self`` and ``other``.

EXAMPLES::

sage: N.<a> = NumberField(x^3-1/2, 'a')

sage: R.<r> = N['r']

sage: f = (5/4*a^2 - 2*a + 4)*r^2 + (5*a^2 - 81/5*a - 17/2)*r + 4/5*a^2 + 24*a + 6

sage: g = (5/4*a^2 - 2*a + 4)*r^2 + (-11*a^2 + 79/5*a - 7/2)*r - 4/5*a^2 - 24*a - 6

sage: gcd(f, g**2)

r - 60808/96625*a^2 - 69936/96625*a - 149212/96625

sage: R = QQ[I]['x']

sage: f = R.random_element(2)

sage: g = f + 1

sage: h = R.random_element(2).monic()

sage: f *=h

sage: g *=h

sage: gcd(f, g) - h

sage: f.gcd(g) - h

TESTS:

Test for degree one extensions::

sage: x = var('x')

sage: N = NumberField(x-3, 'a')

sage: a = N.gen()

sage: R = N['x']

sage: f = R.random_element()

sage: g1 = R.random_element()

sage: g2 = g1*R.random_element() + 1

sage: g1 *= f

sage: g2 *= f

sage: d = gcd(g1, g2)

sage: f.monic() - d

sage: d.parent() is R

True

Test for coercion with other rings and force weird variables

to test PARI behavior::

sage: r = var('r')

sage: N = NumberField(r^2 - 2, 'r')

sage: a = N.gen()

sage: R = N['r']

sage: r = R.gen()

sage: f = N.random_element(4)*r + 1

sage: g = ZZ['r']([1, 2, 3, 4, 5, 6, 7]); g

7*r^6 + 6*r^5 + 5*r^4 + 4*r^3 + 3*r^2 + 2*r + 1

sage: gcd(f, g) == gcd(g, f)

True

sage: h = f.gcd(g); h

sage: h.parent()

Univariate Polynomial Ring in r over Number Field in r with defining polynomial r^2 - 2

sage: gcd([a*r+2, r^2-2])

r + r

"""

if self.is_zero():

if other.is_zero():

return self

else:

return other.monic()

elif other.is_zero():

return self.monic()

elif self.degree() == 0 or other.degree() == 0:

return self.parent().one()

# If the extension is of degree one, use the gcd from QQ[x]

if self.base_ring().degree().is_one():

R = self.base_ring()

a = self.change_ring(QQ)

b = other.change_ring(QQ)

g = a.gcd(b)

return g.change_ring(R)

h1 = self._pari_with_name('x')

h2 = other._pari_with_name('x')

g = h1.gcd(h2)

return (self.parent()(g)).monic()

class Polynomial_relative_number_field_dense(Polynomial_generic_dense_field):

"""

Class of dense univariate polynomials over a relative number field.

"""

def __init__(self, parent, x=None, check=True, is_gen=False, construct=False):

"""

Create a new polynomial in the polynomial ring ``parent``.

INPUT:

- ``parent`` -- polynomial ring in which to construct the

element.

- ``x`` -- (default: None) an object representing the

polynomial, e.g. a list of coefficients. See

:meth:`sage.rings.polynomial.polynomial_element_generic.Polynomial_generic_dense_field.__init__`

for more details.

- ``check`` -- boolean (default: True) if True, make sure that

the coefficients of the polynomial are in the base ring.

- ``is_gen`` -- boolean (default: False) if True, ``x`` is the

distinguished generator of the polynomial ring.

- ``construct`` -- (default: False) boolean, unused.

EXAMPLES::

sage: f = NumberField([x^2-2, x^2-3], 'a')['x'].random_element()

sage: from sage.rings.polynomial.polynomial_number_field import Polynomial_relative_number_field_dense

sage: isinstance(f, Polynomial_relative_number_field_dense)

True

"""

Polynomial_generic_dense_field.__init__(self, parent, x, check, is_gen, construct)

@coerce_binop

def gcd(self, other):

"""

Compute the monic gcd of two polynomials.

Currently, the method checks corner cases in which one of the

polynomials is zero or a constant. Then, computes an absolute

extension and performs the computations there.

INPUT:

- ``other`` -- a polynomial with the same parent as ``self``.

OUTPUT:

- The monic gcd of ``self`` and ``other``.

See :meth:`Polynomial_absolute_number_field_dense.gcd` for

more details.

EXAMPLES::

sage: N = QQ[sqrt(2), sqrt(3)]

sage: s2, s3 = N.gens()

sage: x = polygen(N)

sage: f = x^4 - 5*x^2 +6

sage: g = x^3 + (-2*s2 + s3)*x^2 + (-2*s3*s2 + 2)*x + 2*s3

sage: gcd(f, g)

x^2 + (-sqrt2 + sqrt3)*x - sqrt3*sqrt2

sage: f.gcd(g)

x^2 + (-sqrt2 + sqrt3)*x - sqrt3*sqrt2

TESTS::

sage: x = var('x')

sage: R = NumberField([x^2-2, x^2-3], 'a')['x']

sage: f = R.random_element()

sage: g1 = R.random_element()

sage: g2 = R.random_element()*g1+1

sage: g1 *= f

sage: g2 *= f

sage: f.monic() - g1.gcd(g2)

Test for degree one extensions::

sage: R = NumberField([x-2,x+1,x-3],'a')['x']

sage: f = R.random_element(2)

sage: g1 = R.random_element(2)

sage: g2 = R.random_element(2)*g1+1

sage: g1 *= f

sage: g2 *= f

sage: d = gcd(g1, g2)

sage: d - f.monic()

sage: d.parent() is R

True

Test for hardcoded variables::

sage: R = N['sqrt2sqrt3']

sage: x = R.gen()

sage: f = x^2 - 2

sage: g1 = x^2 - s3

sage: g2 = x - s2

sage: gcd(f, g1)

sage: gcd(f, g2)

sqrt2sqrt3 - sqrt2

"""

if self.is_zero():

if other.is_zero():

return self

else:

return other.monic()

elif other.is_zero():

return self.monic()

elif self.degree() == 0 or other.degree() == 0:

return self.parent().one()

L = self.parent()

x = L.variable_name()

N = self.base_ring()

c = ''.join(map(str,N.variable_names()))

M = N.absolute_field(c)

M_to_N, N_to_M = M.structure()

R = PolynomialRing(M, x)

first = R([N_to_M(foo) for foo in self.list()])

second = R([N_to_M(foo) for foo in other.list()])

result = first.gcd(second)

result = L([M_to_N(foo) for foo in result.list()])

# the result is already monic

return result

Coverage for local/lib/python2.7/site-packages/sage/rings/polynomial/polynomial_number_field.pyx : 94%

52 statements 49 run 3 missing 0 excluded