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

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

Python

>>> 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

Sage Live

K = FractionField(PolynomialRing(QQ, 'x'))
K
loads(K.dumps()) == K
x = K.gen()
f = (x^3 + x)/(17 - x^19); f
loads(f.dumps()) == f

denominator()[source]¶

Return the denominator of self.

EXAMPLES:

Sage

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

Python

>>> 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

Sage Live

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

is_one()[source]¶

Return True if this element is equal to one.

EXAMPLES:

Sage

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

Python

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

Sage Live

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

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:

boolean; whether or not a square
object (optional); an actual square root if found, and None otherwise

EXAMPLES:

Sage

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

Python

>>> 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

Sage Live

R.<t> = QQ[]
(1/t).is_square()
(1/t^6).is_square()
((1+t)^4/t^6).is_square()
(4*(1+t)^4/t^6).is_square()
(2*(1+t)^4/t^6).is_square()
((1+t)/t^6).is_square()
(4*(1+t)^4/t^6).is_square(root=True)
(2*(1+t)^4/t^6).is_square(root=True)
R.<x> = QQ[]
a = 2*(x+1)^2 / (2*(x-1)^2); a
a.is_square()
(0/x).is_square()

is_zero()[source]¶

Return True if this element is equal to zero.

EXAMPLES:

Sage

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

Python

>>> 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

Sage Live

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

nth_root(n)[source]¶

Return a n-th root of this element.

EXAMPLES:

Sage

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

Python

>>> 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

Sage Live

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

numerator()[source]¶

Return the numerator of self.

EXAMPLES:

Sage

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

Python

>>> 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

Sage Live

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

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

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

Python

>>> 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

Sage Live

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

specialization(D=None, phi=None)[source]¶: Return 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

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)

Python

>>> 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)

Sage Live

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

valuation(v=None)[source]¶

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

EXAMPLES:

Sage

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

Python

>>> 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

Sage Live

x = PolynomialRing(RationalField(),'x').gen()
f = (x^3 + x)/(x^2 - 2*x^3)
f
f.valuation()
f.valuation(x^2 + 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]¶

Return whether this element is actually a polynomial.

EXAMPLES:

Sage

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

Python

>>> 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

Sage Live

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

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

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

Python

>>> 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

Sage Live

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

Compare with:

Sage

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

Python

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

Sage Live

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

support()[source]¶

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

EXAMPLES:

Sage

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]

Python

>>> 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 Live

R.<t> = QQ[]
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)
h.support()                                                           # needs sage.libs.pari

sage.rings.fraction_field_element.is_FractionFieldElement(x)[source]¶

Return whether or not x is a FractionFieldElement.

EXAMPLES:

Sage

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

Python

>>> 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 Live

from sage.rings.fraction_field_element import is_FractionFieldElement
R.<x> = ZZ[]
is_FractionFieldElement(x/2)
is_FractionFieldElement(2/x)
is_FractionFieldElement(1/3)

sage.rings.fraction_field_element.make_element(parent, numerator, denominator)[source]¶

Used for unpickling FractionFieldElement objects (and subclasses).

EXAMPLES:

Sage

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)

Python

>>> 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 Live

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

sage.rings.fraction_field_element.make_element_old(parent, cdict)[source]¶

Used for unpickling old FractionFieldElement pickles.

EXAMPLES:

Sage

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)

Python

>>> 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)

Sage Live

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