Fraction fields of Ore polynomial rings.

Sage provides support for building the fraction field of any Ore polynomial ring and performing basic operations in it. The fraction field is constructed by the method sage.rings.polynomial.ore_polynomial_ring.OrePolynomialRing.fraction_field() as demonstrated below:

sage: R.<t> = QQ[]
sage: der = R.derivation()
sage: A.<d> = R['d', der]
sage: K = A.fraction_field()
sage: K
Ore Function Field in d over Fraction Field of Univariate Polynomial Ring in t over Rational Field twisted by d/dt

The simplest way to build elements in \(K\) is to use the division operator over Ore polynomial rings:

sage: f = 1/d
sage: f
d^(-1)
sage: f.parent() is K
True

REPRESENTATION OF ELEMENTS:

Elements in \(K\) are internally represented by fractions of the form \(s^{-1} t\) with the denominator on the left. Notice that, because of noncommutativity, this is not the same that fractions with denominator on the right. For example, a fraction created by the division operator is usually displayed with a different numerator and/or a different denominator:

sage: g = t / d
sage: g
(d - 1/t)^(-1) * t

The left numerator and right denominator are accessible as follows:

sage: g.left_numerator() t sage: g.right_denominator() d

Similarly the methods OrePolynomial.left_denominator() and OrePolynomial.right_numerator() give access to the Ore polynomials \(s\) and \(t\) in the representation \(s^{-1} t\):

sage: g.left_denominator()
d - 1/t
sage: g.right_numerator()
t

We favored the writing \(s^{-1} t\) because it always exists. On the contrary, the writing \(s t^{-1}\) is only guaranteed when the twisting morphism defining the Ore polynomial ring is bijective. As a consequence, when the latter assumption is not fulfilled (or actually if Sage cannot invert the twisting morphism), computing the left numerator and the right denominator fails:

sage: sigma = R.hom([t^2])
sage: S.<x> = R['x', sigma]
sage: F = S.fraction_field()
sage: f = F.random_element()
sage: f.left_numerator()
Traceback (most recent call last):
...
NotImplementedError: inversion of the twisting morphism Ring endomorphism of Fraction Field of Univariate Polynomial Ring in t over Rational Field
  Defn: t |--> t^2

On a related note, fractions are systematically simplified when the twisting morphism is bijective but they are not otherwise. As an example, compare the two following computations:

sage: P = d^2 + t*d + 1
sage: Q = d + t^2
sage: D = d^3 + t^2 + 1
sage: f = P^(-1) * Q
sage: f
(d^2 + t*d + 1)^(-1) * (d + t^2)
sage: g = (D*P)^(-1) * (D*Q)
sage: g
(d^2 + t*d + 1)^(-1) * (d + t^2)

sage: P = x^2 + t*x + 1
sage: Q = x + t^2
sage: D = x^3 + t^2 + 1
sage: f = P^(-1) * Q
sage: f
(x^2 + t*x + 1)^(-1) * (x + t^2)
sage: g = (D*P)^(-1) * (D*Q)
sage: g
(x^5 + t^8*x^4 + x^3 + (t^2 + 1)*x^2 + (t^3 + t)*x + t^2 + 1)^(-1) * (x^4 + t^16*x^3 + (t^2 + 1)*x + t^4 + t^2)
sage: f == g
True

OPERATIONS:

Basic arithmetical operations are available:

sage: f = 1 / d
sage: g = 1 / (d + t)
sage: u = f + g; u
(d^2 + ((t^2 - 1)/t)*d)^(-1) * (2*d + (t^2 - 2)/t)
sage: v = f - g; v
(d^2 + ((t^2 - 1)/t)*d)^(-1) * t
sage: u + v
d^(-1) * 2

sage: f * g
(d^2 + t*d)^(-1)
sage: f / g
d^(-1) * (d + t)

Of course, multiplication remains noncommutative:

sage: g * f
(d^2 + t*d + 1)^(-1)
sage: g^(-1) * f
(d - 1/t)^(-1) * (d + (t^2 - 1)/t)

AUTHOR:

• Xavier Caruso (2020-05)

class sage.rings.polynomial.ore_function_field.OreFunctionCenterInjection(domain, codomain, ringembed)

Bases: sage.rings.morphism.RingHomomorphism

Canonical injection of the center of a Ore function field into this field.

section()

Return a section of this morphism.

EXAMPLES:

sage: k.<a> = GF(5^3)
sage: S.<x> = SkewPolynomialRing(k, k.frobenius_endomorphism())
sage: K = S.fraction_field()
sage: Z = K.center()
sage: iota = K.coerce_map_from(Z)
sage: sigma = iota.section()
sage: sigma(x^3 / (x^6 + 1))
z/(z^2 + 1)

class sage.rings.polynomial.ore_function_field.OreFunctionField(ring, category=None)

Bases: sage.rings.ring.Algebra, sage.structure.unique_representation.UniqueRepresentation

A class for fraction fields of Ore polynomial rings.

change_var(var)

Return the Ore function field in variable var with the same base ring, twisting morphism and twisting derivation as self.

INPUT:

• var – a string representing the name of the new variable.

EXAMPLES:

sage: k.<t> = GF(5^3)
sage: Frob = k.frobenius_endomorphism()
sage: R.<x> = OrePolynomialRing(k,Frob)
sage: K = R.fraction_field()
sage: K
Ore Function Field in x over Finite Field in t of size 5^3 twisted by t |--> t^5

sage: Ky = K.change_var('y'); Ky
Ore Function Field in y over Finite Field in t of size 5^3 twisted by t |--> t^5
sage: Ky is K.change_var('y')
True

characteristic()

Return the characteristic of this Ore function field.

EXAMPLES:

sage: R.<t> = QQ[]
sage: sigma = R.hom([t+1])
sage: S = R['x',sigma]
sage: S.fraction_field().characteristic()
0

sage: k.<u> = GF(5^3)
sage: Frob = k.frobenius_endomorphism()
sage: S = k['y',Frob]
sage: S.fraction_field().characteristic()
5

fraction_field()

Return the fraction field of this Ore function field, i.e. this Ore function field itself.

EXAMPLES:

sage: R.<t> = QQ[]
sage: der = R.derivation()
sage: A.<d> = R['d', der]
sage: K = A.fraction_field()

sage: K
Ore Function Field in d over Fraction Field of Univariate Polynomial Ring in t over Rational Field twisted by d/dt
sage: K.fraction_field()
Ore Function Field in d over Fraction Field of Univariate Polynomial Ring in t over Rational Field twisted by d/dt
sage: K.fraction_field() is K
True

gen(n=0)

Return the indeterminate generator of this Ore function field.

INPUT:

• n – index of generator to return (default: 0). Exists for compatibility with other polynomial rings.

EXAMPLES:

sage: k.<a> = GF(5^4)
sage: Frob = k.frobenius_endomorphism()
sage: S.<x> = k['x', Frob]
sage: K = S.fraction_field()
sage: K.gen()
x

gens_dict()

Return a {name: variable} dictionary of the generators of this Ore function field.

EXAMPLES:

sage: R.<t> = ZZ[]
sage: sigma = R.hom([t+1])
sage: S.<x> = OrePolynomialRing(R, sigma)
sage: K = S.fraction_field()

sage: K.gens_dict()
{'x': x}

is_commutative()

Return True if this Ore function field is commutative, i.e. if the twisting morphism is the identity and the twisting derivation vanishes.

EXAMPLES:

sage: k.<a> = GF(5^3)
sage: Frob = k.frobenius_endomorphism()
sage: S.<x> = k['x', Frob]
sage: K = S.fraction_field()
sage: K.is_commutative()
False

sage: T.<y> = k['y', Frob^3]
sage: L = T.fraction_field()
sage: L.is_commutative()
True

is_exact()

Return True if elements of this Ore function field are exact. This happens if and only if elements of the base ring are exact.

EXAMPLES:

sage: k.<t> = GF(5^3)
sage: Frob = k.frobenius_endomorphism()
sage: S.<x> = k['x', Frob]
sage: K = S.fraction_field()
sage: K.is_exact()
True

sage: k.<u> = Qq(5^3)
sage: Frob = k.frobenius_endomorphism()
sage: S.<x> = k['x', Frob]
sage: K = S.fraction_field()
sage: K.is_exact()
False

is_field()

Return always True since Ore function field are (skew) fields.

EXAMPLES:

sage: k.<a> = GF(5^3)
sage: Frob = k.frobenius_endomorphism()
sage: S.<x> = k['x', Frob]
sage: K = S.fraction_field()

sage: S.is_field()
False
sage: K.is_field()
True

is_finite()

Return False since Ore function field are not finite.

EXAMPLES:

sage: k.<t> = GF(5^3)
sage: k.is_finite()
True
sage: Frob = k.frobenius_endomorphism()
sage: S.<x> = k['x',Frob]
sage: K = S.fraction_field()
sage: K.is_finite()
False

is_sparse()

Return True if the elements of this Ore function field are sparsely represented.

Warning

Since sparse Ore polynomials are not yet implemented, this function always returns False.

EXAMPLES:

sage: R.<t> = RR[]
sage: sigma = R.hom([t+1])
sage: S.<x> = R['x', sigma]
sage: K = S.fraction_field()
sage: K.is_sparse()
False

ngens()

Return the number of generators of this Ore function field, which is \(1\).

EXAMPLES:

sage: R.<t> = RR[]
sage: sigma = R.hom([t+1])
sage: S.<x> = R['x',sigma]
sage: K = S.fraction_field()
sage: K.ngens()
1

parameter(n=0)

Return the indeterminate generator of this Ore function field.

INPUT:

• n – index of generator to return (default: 0). Exists for compatibility with other polynomial rings.

EXAMPLES:

sage: k.<a> = GF(5^4)
sage: Frob = k.frobenius_endomorphism()
sage: S.<x> = k['x', Frob]
sage: K = S.fraction_field()
sage: K.gen()
x

random_element(degree=2, monic=False, *args, **kwds)

Return a random Ore function in this field.

INPUT:

• degree – (default: 2) an integer or a list of two integers; the degrees of the denominator and numerator

• monic – (default: False) if True, return a monic Ore function with monic numerator and denominator

• *args, **kwds – passed in to the random_element method for the base ring

EXAMPLES:

sage: k.<t> = GF(5^3)
sage: Frob = k.frobenius_endomorphism()
sage: S.<x> = k['x', Frob]
sage: K = S.fraction_field()

sage: K.random_element()              # random
(x^2 + (2*t^2 + t + 1)*x + 2*t^2 + 2*t + 3)^(-1) * ((2*t^2 + 3)*x^2 + (4*t^2 + t + 4)*x + 2*t^2 + 2)
sage: K.random_element(monic=True)    # random
(x^2 + (4*t^2 + 3*t + 4)*x + 4*t^2 + t)^(-1) * (x^2 + (2*t^2 + t + 3)*x + 3*t^2 + t + 2)
sage: K.random_element(degree=3)      # random
(x^3 + (2*t^2 + 3)*x^2 + (2*t^2 + 4)*x + t + 3)^(-1) * ((t + 4)*x^3 + (4*t^2 + 2*t + 2)*x^2 + (2*t^2 + 3*t + 3)*x + 3*t^2 + 3*t + 1)
sage: K.random_element(degree=[2,5])  # random
(x^2 + (4*t^2 + 2*t + 2)*x + 4*t^2 + t + 2)^(-1) * ((3*t^2 + t + 1)*x^5 + (2*t^2 + 2*t)*x^4 + (t^2 + 2*t + 4)*x^3 + (3*t^2 + 2*t)*x^2 + (t^2 + t + 4)*x)

twisting_derivation()

Return the twisting derivation defining this Ore function field or None if this Ore function field is not twisted by a derivation.

EXAMPLES:

sage: R.<t> = QQ[]
sage: der = R.derivation(); der
d/dt
sage: A.<d> = R['d', der]
sage: F = A.fraction_field()
sage: F.twisting_derivation()
d/dt

sage: k.<a> = GF(5^3)
sage: Frob = k.frobenius_endomorphism()
sage: S.<x> = k['x', Frob]
sage: K = S.fraction_field()
sage: K.twisting_derivation()

See also

sage.rings.polynomial.ore_polynomial_element.OrePolynomial.twisting_derivation(), twisting_morphism()

twisting_morphism(n=1)

Return the twisting endomorphism defining this Ore function field iterated n times or None if this Ore function field is not twisted by an endomorphism.

INPUT:

• n - an integer (default: 1)

EXAMPLES:

sage: R.<t> = QQ[]
sage: sigma = R.hom([t+1])
sage: S.<x> = R['x', sigma]
sage: K = S.fraction_field()
sage: K.twisting_morphism()
Ring endomorphism of Fraction Field of Univariate Polynomial Ring in t over Rational Field
  Defn: t |--> t + 1

When the Ore polynomial ring is only twisted by a derivation, this method returns nothing:

sage: der = R.derivation()
sage: A.<d> = R['x', der]
sage: F = A.fraction_field()
sage: F.twisting_morphism()

See also

sage.rings.polynomial.ore_polynomial_element.OrePolynomial.twisting_morphism(), twisting_derivation()

class sage.rings.polynomial.ore_function_field.OreFunctionField_with_large_center(ring, category=None)

Bases: sage.rings.polynomial.ore_function_field.OreFunctionField

A specialized class for Ore polynomial fields whose center has finite index.

center(name=None, names=None, default=False)

Return the center of this Ore function field.

Note

One can prove that the center is a field of rational functions over a subfield of the base ring of this Ore function field.

INPUT:

• name – a string or None (default: None); the name for the central variable

• default – a boolean (default: False); if True, set the default variable name for the center to name

EXAMPLES:

sage: k.<t> = GF(5^3)
sage: Frob = k.frobenius_endomorphism()
sage: S.<x> = k['x',Frob]
sage: K = S.fraction_field()

sage: Z = K.center(); Z
Fraction Field of Univariate Polynomial Ring in z over Finite Field of size 5

We can pass in another variable name:

sage: K.center(name='y')
Fraction Field of Univariate Polynomial Ring in y over Finite Field of size 5

or use the bracket notation:

sage: Zy.<y> = K.center(); Zy
Fraction Field of Univariate Polynomial Ring in y over Finite Field of size 5

A coercion map from the center to the Ore function field is set:

sage: K.has_coerce_map_from(Zy)
True

and pushout works:

sage: x.parent()
Ore Polynomial Ring in x over Finite Field in t of size 5^3 twisted by t |--> t^5
sage: y.parent()
Fraction Field of Univariate Polynomial Ring in y over Finite Field of size 5
sage: P = x + y; P
x^3 + x
sage: P.parent()
Ore Function Field in x over Finite Field in t of size 5^3 twisted by t |--> t^5

A conversion map in the reverse direction is also set:

sage: Zy(x^(-6) + 2)
(2*y^2 + 1)/y^2

sage: Zy(1/x^2)
Traceback (most recent call last):
...
ValueError: x^(-2) is not in the center

ABOUT THE DEFAULT NAME OF THE CENTRAL VARIABLE:

A priori, the default is z.

However, a variable name is given the first time this method is called, the given name become the default for the next calls:

sage: k.<t> = GF(11^3)
sage: phi = k.frobenius_endomorphism()
sage: S.<X> = k['X', phi]
sage: K = S.fraction_field()

sage: C.<u> = K.center()  # first call
sage: C
Fraction Field of Univariate Polynomial Ring in u over Finite Field of size 11
sage: K.center()  # second call: the variable name is still u
Fraction Field of Univariate Polynomial Ring in u over Finite Field of size 11

We can update the default variable name by passing in the argument default=True:

sage: D.<v> = K.center(default=True)
sage: D
Fraction Field of Univariate Polynomial Ring in v over Finite Field of size 11
sage: K.center()
Fraction Field of Univariate Polynomial Ring in v over Finite Field of size 11

class sage.rings.polynomial.ore_function_field.SectionOreFunctionCenterInjection(embed)

Bases: sage.categories.map.Section

Section of the canonical injection of the center of a Ore function field into this field