Fraction Field Elements

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 [Ho72]

REFERENCES:

[Ho72]E. Horowitz, “Algorithms for Rational Function Arithmetic Operations”, Annual ACM Symposium on Theory of Computing, Proceedings of the Fourth Annual ACM Symposium on Theory of Computing, pp. 108–118, 1972
class sage.rings.fraction_field_element.FractionFieldElement

Bases: sage.structure.element.FieldElement

EXAMPLES:

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

TESTS:

Test if trac ticket #5451 is fixed:

sage: A = FiniteField(9,'theta')['t']
sage: K.<t> = FractionField(A)
sage: f= 2/(t^2+2*t); g =t^9/(t^18 + t^10 + t^2);f+g
(2*t^15 + 2*t^14 + 2*t^13 + 2*t^12 + 2*t^11 + 2*t^10 + 2*t^9 + t^7 + t^6 + t^5 + t^4 + t^3 + t^2 + t + 1)/(t^17 + t^9 + t)

Test if trac ticket #8671 is fixed:

sage: P.<n> = QQ[]
sage: F = P.fraction_field()
sage: P.one_element()//F.one_element()
1
sage: F.one_element().quo_rem(F.one_element())
(1, 0)
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)

Returns 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
(2*x^2 + 4*x + 2)/(2*x^2 - 4*x + 2)
sage: a.numerator().is_square()
False
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
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()

Divides out the gcd of the numerator and denominator.

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)[]
sage: f = (x^2+2*x+1)/(x+1); f
(x^2 + 2.0*x + 1.0)/(x + 1.0)
sage: f.reduce(); f
x + 1.0
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
(x^2 + 1)/(-2*x^2 + x)
sage: f.valuation()
-1
sage: f.valuation(x^2+1)
1
class sage.rings.fraction_field_element.FractionFieldElement_1poly_field

Bases: sage.rings.fraction_field_element.FractionFieldElement

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

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
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()
[t - 1, t + 3, t^2 + 2, t^2 + t + 1, t^4 - 2]
sage.rings.fraction_field_element.is_FractionFieldElement(x)

Returns whether or not x is a :class`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)

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