Fraction Field Elements#

AUTHORS:

  • William Stein (input from David Joyner, David Kohel, and Joe Wetherell)

  • Sebastian Pancratz (2010-01-06): Rewrite of addition, multiplication and derivative to use Henrici’s algorithms [Hor1972]

class sage.rings.fraction_field_element.FractionFieldElement[source]#

Bases: FieldElement

EXAMPLES:

sage: K = FractionField(PolynomialRing(QQ, 'x'))
sage: K
Fraction Field of Univariate Polynomial Ring in x over Rational Field
sage: loads(K.dumps()) == K
True
sage: x = K.gen()
sage: f = (x^3 + x)/(17 - x^19); f
(-x^3 - x)/(x^19 - 17)
sage: loads(f.dumps()) == f
True

>>> from sage.all import *
>>> K = FractionField(PolynomialRing(QQ, 'x'))
>>> K
Fraction Field of Univariate Polynomial Ring in x over Rational Field
>>> loads(K.dumps()) == K
True
>>> x = K.gen()
>>> f = (x**Integer(3) + x)/(Integer(17) - x**Integer(19)); f
(-x^3 - x)/(x^19 - 17)
>>> loads(f.dumps()) == f
True
denominator()[source]#

Return the denominator of self.

EXAMPLES:

sage: R.<x,y> = ZZ[]
sage: f = x/y + 1; f
(x + y)/y
sage: f.denominator()
y

>>> from sage.all import *
>>> R = ZZ['x, y']; (x, y,) = R._first_ngens(2)
>>> f = x/y + Integer(1); f
(x + y)/y
>>> f.denominator()
y
is_one()[source]#

Return True if this element is equal to one.

EXAMPLES:

sage: F = ZZ['x,y'].fraction_field()
sage: x,y = F.gens()
sage: (x/x).is_one()
True
sage: (x/y).is_one()
False

>>> from sage.all import *
>>> F = ZZ['x,y'].fraction_field()
>>> x,y = F.gens()
>>> (x/x).is_one()
True
>>> (x/y).is_one()
False
is_square(root=False)[source]#

Return whether or not self is a perfect square.

If the optional argument root is True, then also returns a square root (or None, if the fraction field element is not square).

INPUT:

  • root – whether or not to also return a square root (default: False)

OUTPUT:

  • bool – whether or not a square

  • object – (optional) an actual square root if found, and None otherwise.

EXAMPLES:

sage: R.<t> = QQ[]
sage: (1/t).is_square()
False
sage: (1/t^6).is_square()
True
sage: ((1+t)^4/t^6).is_square()
True
sage: (4*(1+t)^4/t^6).is_square()
True
sage: (2*(1+t)^4/t^6).is_square()
False
sage: ((1+t)/t^6).is_square()
False

sage: (4*(1+t)^4/t^6).is_square(root=True)
(True, (2*t^2 + 4*t + 2)/t^3)
sage: (2*(1+t)^4/t^6).is_square(root=True)
(False, None)

sage: R.<x> = QQ[]
sage: a = 2*(x+1)^2 / (2*(x-1)^2); a
(x^2 + 2*x + 1)/(x^2 - 2*x + 1)
sage: a.is_square()
True
sage: (0/x).is_square()
True

>>> from sage.all import *
>>> R = QQ['t']; (t,) = R._first_ngens(1)
>>> (Integer(1)/t).is_square()
False
>>> (Integer(1)/t**Integer(6)).is_square()
True
>>> ((Integer(1)+t)**Integer(4)/t**Integer(6)).is_square()
True
>>> (Integer(4)*(Integer(1)+t)**Integer(4)/t**Integer(6)).is_square()
True
>>> (Integer(2)*(Integer(1)+t)**Integer(4)/t**Integer(6)).is_square()
False
>>> ((Integer(1)+t)/t**Integer(6)).is_square()
False

>>> (Integer(4)*(Integer(1)+t)**Integer(4)/t**Integer(6)).is_square(root=True)
(True, (2*t^2 + 4*t + 2)/t^3)
>>> (Integer(2)*(Integer(1)+t)**Integer(4)/t**Integer(6)).is_square(root=True)
(False, None)

>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> a = Integer(2)*(x+Integer(1))**Integer(2) / (Integer(2)*(x-Integer(1))**Integer(2)); a
(x^2 + 2*x + 1)/(x^2 - 2*x + 1)
>>> a.is_square()
True
>>> (Integer(0)/x).is_square()
True
is_zero()[source]#

Return True if this element is equal to zero.

EXAMPLES:

sage: F = ZZ['x,y'].fraction_field()
sage: x,y = F.gens()
sage: t = F(0)/x
sage: t.is_zero()
True
sage: u = 1/x - 1/x
sage: u.is_zero()
True
sage: u.parent() is F
True

>>> from sage.all import *
>>> F = ZZ['x,y'].fraction_field()
>>> x,y = F.gens()
>>> t = F(Integer(0))/x
>>> t.is_zero()
True
>>> u = Integer(1)/x - Integer(1)/x
>>> u.is_zero()
True
>>> u.parent() is F
True
nth_root(n)[source]#

Return a n-th root of this element.

EXAMPLES:

sage: R = QQ['t'].fraction_field()
sage: t = R.gen()
sage: p = (t+1)^3 / (t^2+t-1)^3
sage: p.nth_root(3)
(t + 1)/(t^2 + t - 1)

sage: p = (t+1) / (t-1)
sage: p.nth_root(2)
Traceback (most recent call last):
...
ValueError: not a 2nd power

>>> from sage.all import *
>>> R = QQ['t'].fraction_field()
>>> t = R.gen()
>>> p = (t+Integer(1))**Integer(3) / (t**Integer(2)+t-Integer(1))**Integer(3)
>>> p.nth_root(Integer(3))
(t + 1)/(t^2 + t - 1)

>>> p = (t+Integer(1)) / (t-Integer(1))
>>> p.nth_root(Integer(2))
Traceback (most recent call last):
...
ValueError: not a 2nd power
numerator()[source]#

Return the numerator of self.

EXAMPLES:

sage: R.<x,y> = ZZ[]
sage: f = x/y + 1; f
(x + y)/y
sage: f.numerator()
x + y

>>> from sage.all import *
>>> R = ZZ['x, y']; (x, y,) = R._first_ngens(2)
>>> f = x/y + Integer(1); f
(x + y)/y
>>> f.numerator()
x + y
reduce()[source]#

Reduce this fraction.

Divides out the gcd of the numerator and denominator. If the denominator becomes a unit, it becomes 1. Additionally, depending on the base ring, the leading coefficients of the numerator and the denominator may be normalized to 1.

Automatically called for exact rings, but because it may be numerically unstable for inexact rings it must be called manually in that case.

EXAMPLES:

sage: R.<x> = RealField(10)[]                                               # needs sage.rings.real_mpfr
sage: f = (x^2+2*x+1)/(x+1); f                                              # needs sage.rings.real_mpfr
(x^2 + 2.0*x + 1.0)/(x + 1.0)
sage: f.reduce(); f                                                         # needs sage.rings.real_mpfr
x + 1.0

>>> from sage.all import *
>>> R = RealField(Integer(10))['x']; (x,) = R._first_ngens(1)# needs sage.rings.real_mpfr
>>> f = (x**Integer(2)+Integer(2)*x+Integer(1))/(x+Integer(1)); f                                              # needs sage.rings.real_mpfr
(x^2 + 2.0*x + 1.0)/(x + 1.0)
>>> f.reduce(); f                                                         # needs sage.rings.real_mpfr
x + 1.0
specialization(D=None, phi=None)[source]#

Returns the specialization of a fraction element of a polynomial ring

subs(in_dict=None, *args, **kwds)[source]#

Substitute variables in the numerator and denominator of self.

If a dictionary is passed, the keys are mapped to generators of the parent ring. Otherwise, the arguments are transmitted unchanged to the method subs of the numerator and the denominator.

EXAMPLES:

sage: x, y = PolynomialRing(ZZ, 2, 'xy').gens()
sage: f = x^2 + y + x^2*y^2 + 5
sage: (1/f).subs(x=5)
1/(25*y^2 + y + 30)

>>> from sage.all import *
>>> x, y = PolynomialRing(ZZ, Integer(2), 'xy').gens()
>>> f = x**Integer(2) + y + x**Integer(2)*y**Integer(2) + Integer(5)
>>> (Integer(1)/f).subs(x=Integer(5))
1/(25*y^2 + y + 30)
valuation(v=None)[source]#

Return the valuation of self, assuming that the numerator and denominator have valuation functions defined on them.

EXAMPLES:

sage: x = PolynomialRing(RationalField(),'x').gen()
sage: f = (x^3 + x)/(x^2 - 2*x^3)
sage: f
(-1/2*x^2 - 1/2)/(x^2 - 1/2*x)
sage: f.valuation()
-1
sage: f.valuation(x^2 + 1)
1

>>> from sage.all import *
>>> x = PolynomialRing(RationalField(),'x').gen()
>>> f = (x**Integer(3) + x)/(x**Integer(2) - Integer(2)*x**Integer(3))
>>> f
(-1/2*x^2 - 1/2)/(x^2 - 1/2*x)
>>> f.valuation()
-1
>>> f.valuation(x**Integer(2) + Integer(1))
1
class sage.rings.fraction_field_element.FractionFieldElement_1poly_field[source]#

Bases: FractionFieldElement

A fraction field element where the parent is the fraction field of a univariate polynomial ring over a field.

Many of the functions here are included for coherence with number fields.

is_integral()[source]#

Returns whether this element is actually a polynomial.

EXAMPLES:

sage: R.<t> = QQ[]
sage: elt = (t^2 + t - 2) / (t + 2); elt # == (t + 2)*(t - 1)/(t + 2)
t - 1
sage: elt.is_integral()
True
sage: elt = (t^2 - t) / (t+2); elt # == t*(t - 1)/(t + 2)
(t^2 - t)/(t + 2)
sage: elt.is_integral()
False

>>> from sage.all import *
>>> R = QQ['t']; (t,) = R._first_ngens(1)
>>> elt = (t**Integer(2) + t - Integer(2)) / (t + Integer(2)); elt # == (t + 2)*(t - 1)/(t + 2)
t - 1
>>> elt.is_integral()
True
>>> elt = (t**Integer(2) - t) / (t+Integer(2)); elt # == t*(t - 1)/(t + 2)
(t^2 - t)/(t + 2)
>>> elt.is_integral()
False
reduce()[source]#

Pick a normalized representation of self.

In particular, for any a == b, after normalization they will have the same numerator and denominator.

EXAMPLES:

For univariate rational functions over a field, we have:

sage: R.<x> = QQ[]
sage: (2 + 2*x) / (4*x) # indirect doctest
(1/2*x + 1/2)/x

>>> from sage.all import *
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> (Integer(2) + Integer(2)*x) / (Integer(4)*x) # indirect doctest
(1/2*x + 1/2)/x

Compare with:

sage: R.<x> = ZZ[]
sage: (2 + 2*x) / (4*x)
(x + 1)/(2*x)

>>> from sage.all import *
>>> R = ZZ['x']; (x,) = R._first_ngens(1)
>>> (Integer(2) + Integer(2)*x) / (Integer(4)*x)
(x + 1)/(2*x)
support()[source]#

Returns a sorted list of primes dividing either the numerator or denominator of this element.

EXAMPLES:

sage: R.<t> = QQ[]
sage: h = (t^14 + 2*t^12 - 4*t^11 - 8*t^9 + 6*t^8 + 12*t^6 - 4*t^5
....:      - 8*t^3 + t^2 + 2)/(t^6 + 6*t^5 + 9*t^4 - 2*t^2 - 12*t - 18)
sage: h.support()                                                           # needs sage.libs.pari
[t - 1, t + 3, t^2 + 2, t^2 + t + 1, t^4 - 2]

>>> from sage.all import *
>>> R = QQ['t']; (t,) = R._first_ngens(1)
>>> h = (t**Integer(14) + Integer(2)*t**Integer(12) - Integer(4)*t**Integer(11) - Integer(8)*t**Integer(9) + Integer(6)*t**Integer(8) + Integer(12)*t**Integer(6) - Integer(4)*t**Integer(5)
...      - Integer(8)*t**Integer(3) + t**Integer(2) + Integer(2))/(t**Integer(6) + Integer(6)*t**Integer(5) + Integer(9)*t**Integer(4) - Integer(2)*t**Integer(2) - Integer(12)*t - Integer(18))
>>> h.support()                                                           # needs sage.libs.pari
[t - 1, t + 3, t^2 + 2, t^2 + t + 1, t^4 - 2]
sage.rings.fraction_field_element.is_FractionFieldElement(x)[source]#

Return whether or not x is a FractionFieldElement.

EXAMPLES:

sage: from sage.rings.fraction_field_element import is_FractionFieldElement
sage: R.<x> = ZZ[]
sage: is_FractionFieldElement(x/2)
doctest:warning...
DeprecationWarning: The function is_FractionFieldElement is deprecated;
use 'isinstance(..., FractionFieldElement)' instead.
See https://github.com/sagemath/sage/issues/38128 for details.
False
sage: is_FractionFieldElement(2/x)
True
sage: is_FractionFieldElement(1/3)
False

>>> from sage.all import *
>>> from sage.rings.fraction_field_element import is_FractionFieldElement
>>> R = ZZ['x']; (x,) = R._first_ngens(1)
>>> is_FractionFieldElement(x/Integer(2))
doctest:warning...
DeprecationWarning: The function is_FractionFieldElement is deprecated;
use 'isinstance(..., FractionFieldElement)' instead.
See https://github.com/sagemath/sage/issues/38128 for details.
False
>>> is_FractionFieldElement(Integer(2)/x)
True
>>> is_FractionFieldElement(Integer(1)/Integer(3))
False
sage.rings.fraction_field_element.make_element(parent, numerator, denominator)[source]#

Used for unpickling FractionFieldElement objects (and subclasses).

EXAMPLES:

sage: from sage.rings.fraction_field_element import make_element
sage: R = ZZ['x,y']
sage: x,y = R.gens()
sage: F = R.fraction_field()
sage: make_element(F, 1 + x, 1 + y)
(x + 1)/(y + 1)

>>> from sage.all import *
>>> from sage.rings.fraction_field_element import make_element
>>> R = ZZ['x,y']
>>> x,y = R.gens()
>>> F = R.fraction_field()
>>> make_element(F, Integer(1) + x, Integer(1) + y)
(x + 1)/(y + 1)
sage.rings.fraction_field_element.make_element_old(parent, cdict)[source]#

Used for unpickling old FractionFieldElement pickles.

EXAMPLES:

sage: from sage.rings.fraction_field_element import make_element_old
sage: R.<x,y> = ZZ[]
sage: F = R.fraction_field()
sage: make_element_old(F, {'_FractionFieldElement__numerator': x + y,
....:                      '_FractionFieldElement__denominator': x - y})
(x + y)/(x - y)

>>> from sage.all import *
>>> from sage.rings.fraction_field_element import make_element_old
>>> R = ZZ['x, y']; (x, y,) = R._first_ngens(2)
>>> F = R.fraction_field()
>>> make_element_old(F, {'_FractionFieldElement__numerator': x + y,
...                      '_FractionFieldElement__denominator': x - y})
(x + y)/(x - y)