r"""
Factorizations
The :class:`Factorization` class provides a structure for holding quite
general lists of objects with integer multiplicities. These may hold
the results of an arithmetic or algebraic factorization, where the
objects may be primes or irreducible polynomials and the
multiplicities are the (non-zero) exponents in the factorization. For
other types of examples, see below.
:class:`Factorization` class objects contain a ``list``, so can be
printed nicely and be manipulated like a list of prime-exponent pairs,
or easily turned into a plain list. For example, we factor the
integer `-45`::
sage: F = factor(-45)
This returns an object of type :class:`Factorization`::
sage: type(F)
<class 'sage.structure.factorization_integer.IntegerFactorization'>
It prints in a nice factored form::
sage: F
-1 * 3^2 * 5
There is an underlying list representation, \emph{which ignores the
unit part}::
sage: list(F)
[(3, 2), (5, 1)]
A :class:`Factorization` is not actually a list::
sage: isinstance(F, list)
False
However, we can access the :class:`Factorization` F itself as if it were a list::
sage: F[0]
(3, 2)
sage: F[1]
(5, 1)
To get at the unit part, use the :meth:`Factorization.unit` function::
sage: F.unit()
-1
All factorizations are immutable, up to ordering with ``sort()`` and
simplifying with ``simplify()``. Thus if you write a function that
returns a cached version of a factorization, you do not have to return
a copy.
::
sage: F = factor(-12); F
-1 * 2^2 * 3
sage: F[0] = (5,4)
Traceback (most recent call last):
...
TypeError: 'Factorization' object does not support item assignment
EXAMPLES:
This more complicated example involving polynomials also illustrates
that the unit part is not discarded from factorizations::
sage: x = QQ['x'].0
sage: f = -5*(x-2)*(x-3)
sage: f
-5*x^2 + 25*x - 30
sage: F = f.factor(); F
(-5) * (x - 3) * (x - 2)
sage: F.unit()
-5
sage: F.value()
-5*x^2 + 25*x - 30
The underlying list is the list of pairs `(p_i, e_i)`, where each
`p_i` is a 'prime' and each `e_i` is an integer. The unit part
is discarded by the list::
sage: list(F)
[(x - 3, 1), (x - 2, 1)]
sage: len(F)
2
sage: F[1]
(x - 2, 1)
In the ring `\ZZ[x]`, the integer `-5` is not a unit, so the
factorization has three factors::
sage: x = ZZ['x'].0
sage: f = -5*(x-2)*(x-3)
sage: f
-5*x^2 + 25*x - 30
sage: F = f.factor(); F
(-1) * 5 * (x - 3) * (x - 2)
sage: F.universe()
Univariate Polynomial Ring in x over Integer Ring
sage: F.unit()
-1
sage: list(F)
[(5, 1), (x - 3, 1), (x - 2, 1)]
sage: F.value()
-5*x^2 + 25*x - 30
sage: len(F)
3
On the other hand, -1 is a unit in `\ZZ`, so it is included in the unit::
sage: x = ZZ['x'].0
sage: f = -1*(x-2)*(x-3)
sage: F = f.factor(); F
(-1) * (x - 3) * (x - 2)
sage: F.unit()
-1
sage: list(F)
[(x - 3, 1), (x - 2, 1)]
Factorizations can involve fairly abstract mathematical objects::
sage: F = ModularSymbols(11,4).factorization()
sage: F
(Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 6 for Gamma_0(11) of weight 4 with sign 0 over Rational Field) *
(Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 6 for Gamma_0(11) of weight 4 with sign 0 over Rational Field) *
(Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 6 for Gamma_0(11) of weight 4 with sign 0 over Rational Field)
sage: type(F)
<class 'sage.structure.factorization.Factorization'>
sage: K.<a> = NumberField(x^2 + 3); K
Number Field in a with defining polynomial x^2 + 3
sage: f = K.factor(15); f
(Fractional ideal (-a))^2 * (Fractional ideal (5))
sage: f.universe()
Monoid of ideals of Number Field in a with defining polynomial x^2 + 3
sage: f.unit()
Fractional ideal (1)
sage: g=K.factor(9); g
(Fractional ideal (-a))^4
sage: f.lcm(g)
(Fractional ideal (-a))^4 * (Fractional ideal (5))
sage: f.gcd(g)
(Fractional ideal (-a))^2
sage: f.is_integral()
True
TESTS::
sage: F = factor(-20); F
-1 * 2^2 * 5
sage: G = loads(dumps(F)); G
-1 * 2^2 * 5
sage: G == F
True
sage: G is F
False
AUTHORS:
- William Stein (2006-01-22): added unit part as suggested by David Kohel.
- William Stein (2008-01-17): wrote much of the documentation and
fixed a couple of bugs.
- Nick Alexander (2008-01-19): added support for non-commuting factors.
- John Cremona (2008-08-22): added division, lcm, gcd, is_integral and
universe functions
"""
from sage.structure.sage_object import SageObject
from sage.structure.sequence import Sequence
from sage.rings.integer import Integer
from sage.misc.all import prod
from sage.misc.cachefunc import cached_method
class Factorization(SageObject):
"""
A formal factorization of an object.
EXAMPLES::
sage: N = 2006
sage: F = N.factor(); F
2 * 17 * 59
sage: F.unit()
1
sage: F = factor(-2006); F
-1 * 2 * 17 * 59
sage: F.unit()
-1
sage: loads(F.dumps()) == F
True
sage: F = Factorization([(x,1/3)])
Traceback (most recent call last):
...
TypeError: exponents of factors must be integers
"""
def __init__(self, x, unit=None, cr=False, sort=True, simplify=True):
"""
Create a :class:`Factorization` object.
INPUT:
- ``x`` - a list of pairs (p, e) with e an integer;
otherwise a TypeError is raised
- ``unit`` - (default: 1) the unit part of the factorization.
- ``cr`` - (default: False) if True, print the factorization with
carriage returns between factors.
- ``sort`` - (default: True) if True, sort the factors by calling
the sort function ``self.sort()`` after creating the factorization
- ``simplify`` - (default: True) if True, remove duplicate
factors from the factorization. See the documentation for
self.simplify.
OUTPUT:
- a Factorization object
EXAMPLES:
We create a factorization with all the default options::
sage: Factorization([(2,3), (5, 1)])
2^3 * 5
We create a factorization with a specified unit part::
sage: Factorization([(2,3), (5, 1)], unit=-1)
-1 * 2^3 * 5
We try to create a factorization but with a string an exponent, which
results in a TypeError::
sage: Factorization([(2,3), (5, 'x')])
Traceback (most recent call last):
...
TypeError: exponents of factors must be integers
We create a factorization that puts newlines after each multiply sign
when printing. This is mainly useful when the primes are large::
sage: Factorization([(2,3), (5, 2)], cr=True)
2^3 *
5^2
Another factorization with newlines and nontrivial unit part, which
appears on a line by itself::
sage: Factorization([(2,3), (5, 2)], cr=True, unit=-2)
-2 *
2^3 *
5^2
A factorization, but where we do not sort the factors::
sage: Factorization([(5,3), (2, 3)], sort=False)
5^3 * 2^3
By default, in the commutative case, factorizations are sorted by the
prime base::
sage: Factorization([(2, 7), (5,2), (2, 5)])
2^12 * 5^2
sage: R.<a,b> = FreeAlgebra(QQ,2)
sage: Factorization([(a,1),(b,1),(a,2)])
a * b * a^2
Autosorting (the default) swaps around the factors below::
sage: F = Factorization([(ZZ^3, 2), (ZZ^2, 5)], cr=True); F
(Ambient free module of rank 2 over the principal ideal domain Integer Ring)^5 *
(Ambient free module of rank 3 over the principal ideal domain Integer Ring)^2
"""
if not isinstance(x, list):
raise TypeError, "x must be a list"
for i in xrange(len(x)):
t=x[i]
if not (isinstance(t, tuple) and len(t) == 2):
raise TypeError, "x must be a list of pairs (p, e) with e an integer"
if not isinstance(t[1],(int, long, Integer)):
try:
x[i]= (t[0], Integer(t[1]))
except TypeError:
raise TypeError, "exponents of factors must be integers"
try:
self.__universe = Sequence(t[0] for t in x).universe()
except TypeError:
self.__universe = None
self.__x = [ (t[0],int(t[1])) for t in x]
if unit is None:
if len(x) > 0:
try:
unit = self.__universe(1)
except (AttributeError, TypeError):
unit = Integer(1)
else:
unit = Integer(1)
self.__unit = unit
self.__cr = cr
if sort and self.is_commutative():
self.sort()
if simplify:
self.simplify()
def __getitem__(self, i):
"""
Return `i^{th}` factor of self.
EXAMPLES::
sage: a = factor(-75); a
-1 * 3 * 5^2
sage: a[0]
(3, 1)
sage: a[1]
(5, 2)
sage: a[-1]
(5, 2)
sage: a[5]
Traceback (most recent call last):
...
IndexError: list index out of range
"""
return self.__x.__getitem__(i)
def __setitem__(self, i, v):
"""
Set the `i^{th}` factor of self.
.. warning::
NOT ALLOWED -- Factorizations are immutable.
EXAMPLES::
sage: a = factor(-75); a
-1 * 3 * 5^2
sage: a[0] = (2,3)
Traceback (most recent call last):
...
TypeError: 'Factorization' object does not support item assignment
"""
raise TypeError, "'Factorization' object does not support item assignment"
def __len__(self):
"""
Return the number of prime factors of self, not counting
the unit part.
EXAMPLES::
sage: len(factor(15))
2
Note that the unit part is not included in the count::
sage: a = factor(-75); a
-1 * 3 * 5^2
sage: len(a)
2
sage: list(a)
[(3, 1), (5, 2)]
sage: len(list(a))
2
"""
return len(self.__x)
def __cmp__(self, other):
"""
Compare self and other. This compares the underlying
lists of self and other, ignoring the unit!
EXAMPLES:
We compare two contrived formal factorizations::
sage: a = Factorization([(2, 7), (5,2), (2, 5)])
sage: b = Factorization([(2, 7), (5,10), (7, 3)])
sage: a
2^12 * 5^2
sage: b
2^7 * 5^10 * 7^3
sage: a < b
True
sage: b < a
False
sage: a.value()
102400
sage: b.value()
428750000000
We compare factorizations of some polynomials::
sage: x = polygen(QQ)
sage: x^2 - 1 > x^2 - 4
True
sage: factor(x^2 - 1) > factor(x^2 - 4)
True
"""
if not isinstance(other, Factorization):
return cmp(type(self), type(other))
try:
return cmp(self.value(), other.value())
except Exception:
c = cmp(self.__unit, other.__unit)
if c: return c
return list.__cmp__(self, other)
def __copy__(self):
r"""
Return a copy of self.
This is *not* a deepcopy -- only references to the factors are
returned, not copies of them. Use ``deepcopy(self)`` if you need
a deep copy of self.
EXAMPLES:
We create a factorization that has mutable primes::
sage: F = Factorization([([1,2], 5), ([5,6], 10)]); F
([1, 2])^5 * ([5, 6])^10
We make a copy of it::
sage: G = copy(F); G
([1, 2])^5 * ([5, 6])^10
sage: G is F
False
Note that if we change one of the mutable "primes" of F, this does
change G::
sage: F[1][0][0] = 'hello'
sage: G
([1, 2])^5 * (['hello', 6])^10
"""
return Factorization(self.__x, unit=self.__unit, cr=self.__cr,
sort=False, simplify=False)
def __deepcopy__(self, memo):
r"""
Return a deep copy of self.
EXAMPLES:
We make a factorization that has mutable entries::
sage: F = Factorization([([1,2], 5), ([5,6], 10)]); F
([1, 2])^5 * ([5, 6])^10
Now we make a copy of it and a deep copy::
sage: K = copy(F)
sage: G = deepcopy(F); G
([1, 2])^5 * ([5, 6])^10
We change one of the mutable entries of F::
sage: F[0][0][0] = 10
This of course changes F::
sage: F
([10, 2])^5 * ([5, 6])^10
It also changes the copy K of F::
sage: K
([10, 2])^5 * ([5, 6])^10
It does *not* change the deep copy G::
sage: G
([1, 2])^5 * ([5, 6])^10
"""
import copy
return Factorization(copy.deepcopy(list(self), memo),
cr=self.__cr, sort=False, simplify=False)
def universe(self):
r"""
Return the parent structure of my factors.
.. note::
This used to be called ``base_ring``, but the universe
of a factorization need not be a ring.
EXAMPLES::
sage: F = factor(2006)
sage: F.universe()
Integer Ring
sage: R.<x,y,z> = FreeAlgebra(QQ, 3)
sage: F = Factorization([(z, 2)], 3)
sage: (F*F^-1).universe()
Free Algebra on 3 generators (x, y, z) over Rational Field
sage: F = ModularSymbols(11,4).factorization()
sage: F.universe()
"""
try:
return self.__universe
except AttributeError:
return None
def base_change(self, U):
"""
Return the factorization self, with its factors (including the
unit part) coerced into the universe `U`.
EXAMPLES::
sage: F = factor(2006)
sage: F.universe()
Integer Ring
sage: P.<x> = ZZ[]
sage: F.base_change(P).universe()
Univariate Polynomial Ring in x over Integer Ring
This method will return a TypeError if the coercion is not
possible::
sage: g = x^2 - 1
sage: F = factor(g); F
(x - 1) * (x + 1)
sage: F.universe()
Univariate Polynomial Ring in x over Integer Ring
sage: F.base_change(ZZ)
Traceback (most recent call last):
...
TypeError: Impossible to coerce the factors of (x - 1) * (x + 1) into Integer Ring
"""
if len(self) == 0:
return self
try:
return Factorization([(U(f[0]), f[1]) for f in list(self)], unit=U(self.unit()))
except TypeError:
raise TypeError, "Impossible to coerce the factors of %s into %s"%(self, U)
def is_commutative(self):
"""
Return True if my factors commute.
EXAMPLES::
sage: F = factor(2006)
sage: F.is_commutative()
True
sage: K = QuadraticField(23, 'a')
sage: F = K.factor(13)
sage: F.is_commutative()
True
sage: R.<x,y,z> = FreeAlgebra(QQ, 3)
sage: F = Factorization([(z, 2)], 3)
sage: F.is_commutative()
False
sage: (F*F^-1).is_commutative()
False
"""
try:
return self.universe().is_commutative()
except Exception:
return True
def _set_cr(self, cr):
"""
Change whether or not the factorization is printed with
carriage returns after each factor.
EXAMPLES::
sage: x = polygen(QQ,'x')
sage: F = factor(x^6 - 1); F
(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)
sage: F._set_cr(True); F
(x - 1) *
(x + 1) *
(x^2 - x + 1) *
(x^2 + x + 1)
sage: F._set_cr(False); F
(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)
"""
self.__cr = bool(cr)
def simplify(self):
"""
Combine adjacent products as much as possible.
TESTS::
sage: R.<x,y> = FreeAlgebra(ZZ, 2)
sage: F = Factorization([(x,3), (y, 2), (y,2)], simplify=False); F
x^3 * y^2 * y^2
sage: F.simplify(); F
x^3 * y^4
sage: F * Factorization([(y, -2)], 2)
(2) * x^3 * y^2
"""
repeat = False
simp = []
import itertools
for obj, agroup in itertools.groupby(list(self), lambda x: x[0]):
xs = list(agroup)
if len(xs) > 1:
repeat = True
n = sum([x[1] for x in xs])
if n != 0:
simp.append((obj, n))
self.__x[0:] = simp
if repeat:
self.simplify()
def sort(self, _cmp=None):
r"""
Sort the factors in this factorization.
INPUT:
- ``_cmp`` - (default: None) comparison function
OUTPUT:
- changes this factorization to be sorted
If _cmp is None, we determine the comparison function as
follows: If the prime in the first factor has a dimension
method, then we sort based first on *dimension* then on
the exponent. If there is no dimension method, we next
attempt to sort based on a degree method, in which case, we
sort based first on *degree*, then exponent to break ties
when two factors have the same degree, and if those match
break ties based on the actual prime itself. If there is no
degree method, we sort based on dimension.
EXAMPLES:
We create a factored polynomial::
sage: x = polygen(QQ,'x')
sage: F = factor(x^3 + 1); F
(x + 1) * (x^2 - x + 1)
Then we sort it but using the negated version of the standard
Python cmp function::
sage: F.sort(_cmp = lambda x,y: -cmp(x,y))
sage: F
(x^2 - x + 1) * (x + 1)
"""
if len(self) == 0:
return
if _cmp is None:
try:
a = self.__x[0][0].dimension()
def _cmp(f,g):
"""
This is used internally for comparing. (indirect doctest)
EXAMPLES::
sage: factor(6)
2 * 3
"""
try:
return cmp((f[0].dimension(), f[1]), (g[0].dimension(),g[1]))
except (AttributeError, NotImplementedError):
return cmp((f[0],f[1]), (g[0], g[1]))
except (AttributeError, NotImplementedError):
try:
a = self.__x[0][0].degree()
def _cmp(f,g):
"""
This is used internally for comparing. (indirect doctest)
EXAMPLES::
sage: factor(6)
2 * 3
"""
try:
return cmp((f[0].degree(),f[1],f[0]), (g[0].degree(),g[1],g[0]))
except (AttributeError, NotImplementedError):
return cmp(f[0], g[0])
except (AttributeError, NotImplementedError, TypeError):
self.__x.sort()
return
self.__x.sort(_cmp)
def unit(self):
r"""
Return the unit part of this factorization.
EXAMPLES:
We create a polynomial over the real double field and factor it::
sage: x = polygen(RDF, 'x')
sage: F = factor(-2*x^2 - 1); F
(-2.0) * (x^2 + 0.5)
Note that the unit part of the factorization is `-2.0`::
sage: F.unit()
-2.0
sage: F = factor(-2006); F
-1 * 2 * 17 * 59
sage: F.unit()
-1
"""
return self.__unit
def _cr(self):
"""
Return whether or not factorizations are printed with carriage
returns between factors.
EXAMPLES:
Our first example involves factoring an integer::
sage: F = factor(-93930); F
-1 * 2 * 3 * 5 * 31 * 101
sage: F._cr()
False
sage: F._set_cr(True)
sage: F._cr()
True
This of course looks funny::
sage: F
-1 *
2 *
3 *
5 *
31 *
101
Next we factor a modular symbols space::
sage: F = ModularSymbols(11).factor(); F
(Modular Symbols subspace of dimension 1 of ...) *
(Modular Symbols subspace of dimension 1 of ...) *
(Modular Symbols subspace of dimension 1 of ...)
"""
try:
return self.__cr
except AttributeError:
self.__cr = False
return False
def _repr_(self):
"""
Return the string representation of this factorization.
EXAMPLES::
sage: f = factor(-100); f
-1 * 2^2 * 5^2
sage: f._repr_()
'-1 * 2^2 * 5^2'
Note that the default printing of a factorization can be overloaded
using the rename method::
sage: f.rename('factorization of -100')
sage: f
factorization of -100
However _repr_ always prints normally::
sage: f._repr_()
'-1 * 2^2 * 5^2'
EXAMPLES::
sage: x = polygen(QQ)
sage: Factorization([(x-1,1), (x-2,2)])
(x - 1) * (x - 2)^2
sage: Factorization([(x + 1, -3)])
(x + 1)^-3
"""
cr = self._cr()
if len(self) == 0:
return repr(self.__unit)
s = ''
mul = ' * '
if cr:
mul += '\n'
x = self.__x[0][0]
try:
atomic = (isinstance(x, (int, long)) or
self.universe()._repr_option('element_is_atomic'))
except AttributeError:
atomic = False
if hasattr(x, 'parent'):
one = x.parent()(1)
else:
one = 1
for i in range(len(self)):
t = repr(self.__x[i][0])
n = self.__x[i][1]
if (n != 1 or len(self) > 1 or self.__unit != one) and not atomic \
and ('+' in t or '-' in t or ' ' in t):
t = '(%s)'%t
if n != 1:
t += '^%s'%n
s += t
if i < len(self)-1:
s += mul
if self.__unit != one:
if atomic:
u = repr(self.__unit)
else:
u = '(%s)'%self.__unit
s = u + mul + s
return s
def _latex_(self):
r"""
Return the LaTeX representation of this factorization.
EXAMPLES::
sage: f = factor(-100); f
-1 * 2^2 * 5^2
sage: latex(f)
-1 \cdot 2^{2} \cdot 5^{2}
sage: f._latex_()
'-1 \\cdot 2^{2} \\cdot 5^{2}'
sage: x = AA['x'].0; factor(x^2 + x + 1)._latex_() # trac 12178
'(x^{2} + x + 1.000000000000000?)'
"""
if len(self) == 0:
return self.__unit._latex_()
try:
atomic = (isinstance(self.__x[0][0], (int, long)) or
self.universe()._repr_option('element_is_atomic'))
except AttributeError:
atomic = False
s = ''
for i in range(len(self)):
t = self.__x[i][0]._latex_()
if not atomic and ('+' in t or '-' in t or ' ' in t):
t = '(%s)'%t
n = self.__x[i][1]
if n != 1:
t += '^{%s}'%n
s += t
if i < len(self)-1:
s += ' \\cdot '
if self.__unit != 1:
if atomic:
u = self.__unit._latex_()
else:
u = '\\left(%s\\right)'%self.__unit._latex_()
s = u + ' \\cdot ' + s
return s
@cached_method
def _pari_(self):
"""
Return the PARI factorization matrix corresponding to ``self``.
EXAMPLES::
sage: f = factor(-24)
sage: pari(f)
[-1, 1; 2, 3; 3, 1]
sage: R.<x> = QQ[]
sage: g = factor(x^10 - 1)
sage: pari(g)
[x - 1, 1; x + 1, 1; x^4 - x^3 + x^2 - x + 1, 1; x^4 + x^3 + x^2 + x + 1, 1]
"""
from sage.libs.pari.all import pari
from itertools import chain
n = len(self)
if self.__unit == 1:
init = ()
else:
init = (self.__unit, 1)
n += 1
entries = init + tuple(chain.from_iterable(self))
return pari.matrix(n, 2, entries)
def __add__(self, other):
"""
Return the (unfactored) sum of self and other.
EXAMPLES::
sage: factor(-10) + 16
6
sage: factor(10) - 16
-6
sage: factor(100) + factor(19)
119
"""
if isinstance(other, Factorization):
other = other.value()
return self.value() + other
def __sub__(self, other):
"""
Return the (unfactored) difference of self and other.
EXAMPLES::
sage: factor(-10) + 16
6
sage: factor(10) - 16
-6
"""
if isinstance(other, Factorization):
other = other.value()
return self.value() - other
def __radd__(self, left):
"""
Return the (unfactored) sum of self and left.
EXAMPLES::
sage: 16 + factor(-10)
6
"""
return self.value() + left
def __rsub__(self, left):
"""
Return the (unfactored) difference of left and self.
EXAMPLES::
sage: 16 - factor(10)
6
"""
return left - self.value()
def __neg__(self):
"""
Return negative of this factorization.
EXAMPLES::
sage: a = factor(-75); a
-1 * 3 * 5^2
sage: -a
3 * 5^2
sage: (-a).unit()
1
"""
unit = -self.__unit
return Factorization(list(self), unit, self.__cr,
sort=False, simplify=False)
def __rmul__(self, left):
"""
Return the product left * self, where left is not a Factorization.
EXAMPLES::
sage: a = factor(15); a
3 * 5
sage: -2 * a
-2 * 3 * 5
sage: a * -2
-2 * 3 * 5
sage: R.<x,y> = FreeAlgebra(QQ,2)
sage: f = Factorization([(x,2),(y,3)]); f
x^2 * y^3
sage: x * f
x^3 * y^3
sage: f * x
x^2 * y^3 * x
Note that this does not automatically factor ``left``::
sage: F = Factorization([(5,3), (2,3)])
sage: 46 * F
2^3 * 5^3 * 46
"""
return Factorization([(left, 1)]) * self
def __mul__(self, other):
r"""
Return the product of two factorizations, which is obtained by
combining together like factors.
If the two factorizations have different universes, this
method will attempt to find a common universe for the
product. A TypeError is raised if this is impossible.
EXAMPLES::
sage: factor(-10) * factor(-16)
2^5 * 5
sage: factor(-10) * factor(16)
-1 * 2^5 * 5
sage: R.<x,y> = FreeAlgebra(ZZ, 2)
sage: F = Factorization([(x,3), (y, 2), (x,1)]); F
x^3 * y^2 * x
sage: F*F
x^3 * y^2 * x^4 * y^2 * x
sage: -1 * F
(-1) * x^3 * y^2 * x
sage: P.<x> = ZZ[]
sage: f = 2*x + 2
sage: c = f.content(); g = f//c
sage: Fc = factor(c); Fc.universe()
Integer Ring
sage: Fg = factor(g); Fg.universe()
Univariate Polynomial Ring in x over Integer Ring
sage: F = Fc * Fg; F.universe()
Univariate Polynomial Ring in x over Integer Ring
sage: [type(a[0]) for a in F]
[<type 'sage.rings.polynomial.polynomial_integer_dense_flint.Polynomial_integer_dense_flint'>,
<type 'sage.rings.polynomial.polynomial_integer_dense_flint.Polynomial_integer_dense_flint'>]
"""
if not isinstance(other, Factorization):
return self * Factorization([(other, 1)])
if len(self) and len(other):
try:
U = Sequence([self[0][0], other[0][0]]).universe()
self = self.base_change(U)
other = other.base_change(U)
except TypeError:
raise TypeError, "Cannot multiply %s and %s because they cannot be coerced into a common universe"%(self,other)
if self.is_commutative() and other.is_commutative():
d1 = dict(self)
d2 = dict(other)
s = {}
for a in set(d1.keys()).union(set(d2.keys())):
s[a] = d1.get(a,0) + d2.get(a,0)
return Factorization(list(s.iteritems()), unit=self.unit()*other.unit())
else:
return Factorization(list(self) + list(other), unit=self.unit()*other.unit())
def __pow__(self, n):
"""
Return the `n^{th}` power of a factorization, which is got by
combining together like factors.
EXAMPLES::
sage: f = factor(-100); f
-1 * 2^2 * 5^2
sage: f^3
-1 * 2^6 * 5^6
sage: f^4
2^8 * 5^8
sage: K.<a> = NumberField(x^3 - 39*x - 91)
sage: F = K.factor(7); F
(Fractional ideal (7, a)) * (Fractional ideal (7, a + 2)) * (Fractional ideal (7, a - 2))
sage: F^9
(Fractional ideal (7, a))^9 * (Fractional ideal (7, a + 2))^9 * (Fractional ideal (7, a - 2))^9
sage: R.<x,y> = FreeAlgebra(ZZ, 2)
sage: F = Factorization([(x,3), (y, 2), (x,1)]); F
x^3 * y^2 * x
sage: F**2
x^3 * y^2 * x^4 * y^2 * x
"""
if not isinstance(n, Integer):
try:
n = Integer(n)
except TypeError:
raise TypeError, "Exponent n (= %s) must be an integer." % n
if n == 1:
return self
if n == 0:
return Factorization([])
if self.is_commutative():
return Factorization([(p, n*e) for p, e in self], unit=self.unit()**n, cr=self.__cr, sort=False, simplify=False)
from sage.groups.generic import power
return power(self, n, Factorization([]))
def __invert__(self):
r"""
Return the formal inverse of the factors in the factorization.
EXAMPLES::
sage: F = factor(2006); F
2 * 17 * 59
sage: F^-1
2^-1 * 17^-1 * 59^-1
sage: R.<x,y> = FreeAlgebra(QQ, 2)
sage: F = Factorization([(x,3), (y, 2), (x,1)], 2); F
(2) * x^3 * y^2 * x
sage: F^-1
(1/2) * x^-1 * y^-2 * x^-3
"""
return Factorization([(p,-e) for p,e in reversed(self)],
cr=self._cr(), unit=self.unit()**(-1))
def __div__(self, other):
r"""
Return the quotient of two factorizations, which is obtained by
multiplying the first by the inverse of the second.
EXAMPLES::
sage: factor(-10) / factor(-16)
2^-3 * 5
sage: factor(-10) / factor(16)
-1 * 2^-3 * 5
sage: R.<x,y> = FreeAlgebra(QQ, 2)
sage: F = Factorization([(x,3), (y, 2), (x,1)]); F
x^3 * y^2 * x
sage: G = Factorization([(y, 1), (x,1)],1); G
y * x
sage: F / G
x^3 * y
"""
if not isinstance(other, Factorization):
return self / Factorization([(other, 1)])
return self * other**-1
def value(self):
"""
Return the product of the factors in the factorization, multiplied out.
EXAMPLES::
sage: F = factor(-2006); F
-1 * 2 * 17 * 59
sage: F.value()
-2006
sage: R.<x,y> = FreeAlgebra(ZZ, 2)
sage: F = Factorization([(x,3), (y, 2), (x,1)]); F
x^3 * y^2 * x
sage: F.value()
x^3*y^2*x
"""
return prod([p**e for p,e in self.__x], self.__unit)
expand = value
prod = value
def gcd(self, other):
r"""
Return the gcd of two factorizations.
If the two factorizations have different universes, this
method will attempt to find a common universe for the
gcd. A TypeError is raised if this is impossible.
EXAMPLES::
sage: factor(-30).gcd(factor(-160))
2 * 5
sage: factor(gcd(-30,160))
2 * 5
sage: R.<x> = ZZ[]
sage: (factor(-20).gcd(factor(5*x+10))).universe()
Univariate Polynomial Ring in x over Integer Ring
"""
if not isinstance(other, Factorization):
raise NotImplementedError, "can't take gcd of factorization and non-factorization"
if len(self) and len(other):
try:
U = Sequence([self[0][0], other[0][0]]).universe()
self = self.base_change(U)
other = other.base_change(U)
except TypeError:
raise TypeError, "Cannot take the gcd of %s and %s because they cannot be coerced into a common universe"%(self,other)
if self.is_commutative() and other.is_commutative():
d1 = dict(self)
d2 = dict(other)
s = {}
for a in set(d1.keys()).intersection(set(d2.keys())):
s[a] = min(d1[a],d2[a])
return Factorization(list(s.iteritems()))
else:
raise NotImplementedError, "gcd is not implemented for non-commutative factorizations"
def lcm(self, other):
r"""
Return the lcm of two factorizations.
If the two factorizations have different universes, this
method will attempt to find a common universe for the
lcm. A TypeError is raised if this is impossible.
EXAMPLES::
sage: factor(-10).lcm(factor(-16))
2^4 * 5
sage: factor(lcm(-10,16))
2^4 * 5
sage: R.<x> = ZZ[]
sage: (factor(-20).lcm(factor(5*x+10))).universe()
Univariate Polynomial Ring in x over Integer Ring
"""
if not isinstance(other, Factorization):
raise NotImplementedError, "can't take lcm of factorization and non-factorization"
if len(self) and len(other):
try:
U = Sequence([self[0][0], other[0][0]]).universe()
self = self.base_change(U)
other = other.base_change(U)
except TypeError:
raise TypeError, "Cannot take the lcm of %s and %s because they cannot be coerced into a common universe"%(self,other)
if self.is_commutative() and other.is_commutative():
d1 = dict(self)
d2 = dict(other)
s = {}
for a in set(d1.keys()).union(set(d2.keys())):
s[a] = max(d1.get(a,0),d2.get(a,0))
return Factorization(list(s.iteritems()))
else:
raise NotImplementedError, "lcm is not implemented for non-commutative factorizations"
def is_integral(self):
r"""
Return True iff all exponents of this Factorization are non-negative.
EXAMPLES::
sage: F = factor(-10); F
-1 * 2 * 5
sage: F.is_integral()
True
sage: F = factor(-10) / factor(16); F
-1 * 2^-3 * 5
sage: F.is_integral()
False
"""
return all([e >=0 for p,e in self.__x])
def radical(self):
"""
Return the factorization of the radical of the value of self.
First, check that all exponents in the factorization are
positive, raise ValueError otherwise. If all exponents are
positive, return self with all exponents set to 1 and with the
unit set to 1.
EXAMPLES::
sage: F = factor(-100); F
-1 * 2^2 * 5^2
sage: F.radical()
2 * 5
sage: factor(1/2).radical()
Traceback (most recent call last):
...
ValueError: All exponents in the factorization must be positive.
"""
if not all([e > 0 for p,e in self.__x]):
raise ValueError, "All exponents in the factorization must be positive."
return Factorization([(p,1) for p,e in self.__x], unit=self.unit().parent()(1), cr=self.__cr, sort=False, simplify=False)
def radical_value(self):
"""
Return the product of the prime factors in self.
First, check that all exponents in the factorization are
positive, raise ValueError otherwise. If all exponents are
positive, return the product of the prime factors in self.
This should be functionally equivalent to
self.radical().value()
EXAMPLES::
sage: F = factor(-100); F
-1 * 2^2 * 5^2
sage: F.radical_value()
10
sage: factor(1/2).radical_value()
Traceback (most recent call last):
...
ValueError: All exponents in the factorization must be positive.
"""
if not all([e > 0 for p,e in self.__x]):
raise ValueError, "All exponents in the factorization must be positive."
return prod([p for p,e in self.__x])
