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

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

denominator()

Return the denominator of self.

EXAMPLES:

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

is_one()

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

is_square(root=False)

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

is_zero()

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

nth_root(n)

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

numerator()

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

reduce()

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

specialization(D=None, phi=None)

Returns the specialization of a fraction element of a polynomial ring

subs(in_dict=None, *args, **kwds)

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)

valuation(v=None)

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

class sage.rings.fraction_field_element.FractionFieldElement_1poly_field

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()

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

reduce()

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

Compare with:

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

support()

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]

sage.rings.fraction_field_element.is_FractionFieldElement(x)

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)
False
sage: is_FractionFieldElement(2/x)
True
sage: is_FractionFieldElement(1/3)
False

sage.rings.fraction_field_element.make_element(parent, numerator, denominator)

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)

sage.rings.fraction_field_element.make_element_old(parent, cdict)

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)