Factorizations

The 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.

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 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, which ignores the unit part:

sage: list(F)
[(3, 2), (5, 1)]

A Factorization is not actually a list:

sage: isinstance(F, list)
False

However, we can access the 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 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: # needs sage.libs.pari
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: # needs sage.libs.pari
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: # needs sage.libs.pari
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: # needs sage.libs.pari
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: # needs sage.modular
sage: F = ModularSymbols(11,4).factorization(); 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: # needs sage.rings.number_field
sage: x = ZZ['x'].0
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 (1/2*a + 3/2))^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 (1/2*a + 3/2))^4
sage: f.lcm(g)
(Fractional ideal (1/2*a + 3/2))^4 * (Fractional ideal (5))
sage: f.gcd(g)
(Fractional ideal (1/2*a + 3/2))^2
sage: f.is_integral()
True

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

class sage.structure.factorization.Factorization(x, unit=None, cr=False, sort=True, simplify=True)

Bases: 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)])                                             # needs sage.symbolic
Traceback (most recent call last):
...
TypeError: no conversion of this rational to integer

base_change(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                                                      # needs sage.libs.pari
(x - 1) * (x + 1)
sage: F.universe()                                                          # needs sage.libs.pari
Univariate Polynomial Ring in x over Integer Ring
sage: F.base_change(ZZ)                                                     # needs sage.libs.pari
Traceback (most recent call last):
...
TypeError: Impossible to coerce the factors of (x - 1) * (x + 1) into Integer Ring

expand()

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)                                          # needs sage.combinat sage.modules
sage: F = Factorization([(x,3), (y, 2), (x,1)]); F                          # needs sage.combinat sage.modules
x^3 * y^2 * x
sage: F.value()                                                             # needs sage.combinat sage.modules
x^3*y^2*x

gcd(other)

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()                          # needs sage.libs.pari
Univariate Polynomial Ring in x over Integer Ring

is_commutative()

Return whether the factors commute.

EXAMPLES:

sage: F = factor(2006)
sage: F.is_commutative()
True

sage: # needs sage.rings.number_field
sage: K = QuadraticField(23, 'a')
sage: F = K.factor(13)
sage: F.is_commutative()
True

sage: # needs sage.combinat sage.modules
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

is_integral()

Return whether 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

lcm(other)

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()                        # needs sage.libs.pari
Univariate Polynomial Ring in x over Integer Ring

prod()

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)                                          # needs sage.combinat sage.modules
sage: F = Factorization([(x,3), (y, 2), (x,1)]); F                          # needs sage.combinat sage.modules
x^3 * y^2 * x
sage: F.value()                                                             # needs sage.combinat sage.modules
x^3*y^2*x

radical()

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

radical_value()

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

simplify()

Combine adjacent products as much as possible.

sort(key=None)

Sort the factors in this factorization.

INPUT:

• key – (default: None); comparison key

OUTPUT:

• changes this factorization to be sorted (inplace)

If key is None, we determine the comparison key 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.

Otherwise, we sort according to the prime itself.

EXAMPLES:

We create a factored polynomial:

sage: x = polygen(QQ, 'x')
sage: F = factor(x^3 + 1); F                                                # needs sage.libs.pari
(x + 1) * (x^2 - x + 1)

We sort it by decreasing degree:

sage: F.sort(key=lambda x: (-x[0].degree(), x))                             # needs sage.libs.pari
sage: F                                                                     # needs sage.libs.pari
(x^2 - x + 1) * (x + 1)

subs(*args, **kwds)

Implement the substitution.

This is assuming that each term can be substituted.

There is another mechanism for substitution in symbolic products.

EXAMPLES:

sage: # needs sage.combinat sage.modules
sage: R.<x,y> = FreeAlgebra(QQ, 2)
sage: F = Factorization([(x,3), (y, 2), (x,1)])
sage: F(x=4)
(1) * 4^3 * y^2 * 4
sage: F.subs({y:2})
x^3 * 2^2 * x

sage: R.<x,y> = PolynomialRing(QQ, 2)
sage: F = Factorization([(x,3), (y, 2), (x,1)])
sage: F(x=4)
4 * 4^3 * y^2
sage: F.subs({y:x})
x * x^2 * x^3
sage: F(x=y+x)
(x + y) * y^2 * (x + y)^3

unit()

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                                             # needs numpy
(-2.0) * (x^2 + 0.5000000000000001)

Note that the unit part of the factorization is \(-2.0\):

sage: F.unit()                                                              # needs numpy
-2.0

sage: F = factor(-2006); F
-1 * 2 * 17 * 59
sage: F.unit()
-1

universe()

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)                                        # needs sage.combinat sage.modules
sage: F = Factorization([(z, 2)], 3)                                        # needs sage.combinat sage.modules
sage: (F*F^-1).universe()                                                   # needs sage.combinat sage.modules
Free Algebra on 3 generators (x, y, z) over Rational Field

sage: F = ModularSymbols(11,4).factorization()                              # needs sage.modular
sage: F.universe()                                                          # needs sage.modular

value()

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)                                          # needs sage.combinat sage.modules
sage: F = Factorization([(x,3), (y, 2), (x,1)]); F                          # needs sage.combinat sage.modules
x^3 * y^2 * x
sage: F.value()                                                             # needs sage.combinat sage.modules
x^3*y^2*x