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: # needs sage.rings.function_field
sage: sigma = R.hom([t^2])
sage: S.<x> = R['x', sigma]
sage: F = S.fraction_field()
sage: f = F.random_element()
sage: while not f:
....:     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: # needs sage.rings.function_field
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: # needs sage.rings.function_field
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: # needs sage.rings.function_field
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: # needs sage.rings.function_field
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: RingHomomorphism

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

section()

Return a section of this morphism.

EXAMPLES:

sage: # needs sage.rings.finite_rings
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: Parent, UniqueRepresentation

A class for fraction fields of Ore polynomial rings.

Element = None

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 – string representing the name of the new variable

EXAMPLES:

sage: # needs sage.rings.finite_rings
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()                                   # needs sage.rings.function_field
0

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

Return the tuple of generators of self.

EXAMPLES:

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

gens_dict()

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

EXAMPLES:

sage: # needs sage.rings.finite_rings
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_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: # needs sage.rings.finite_rings
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: # needs sage.rings.padics
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(proof=False)

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

EXAMPLES:

sage: # needs sage.rings.finite_rings
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: # needs sage.rings.finite_rings
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: # needs sage.rings.function_field sage.rings.real_mpfr
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: # needs sage.rings.function_field sage.rings.real_mpfr
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)

alias of gen().

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 – boolean (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: # needs sage.rings.finite_rings
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: # needs sage.rings.finite_rings
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 – 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()                                                # needs sage.rings.function_field
sage: K.twisting_morphism()                                                 # needs sage.rings.function_field
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()                                                # needs sage.rings.function_field
sage: F.twisting_morphism()                                                 # needs sage.rings.function_field

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

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

EXAMPLES:

sage: # needs sage.rings.finite_rings
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')                                                    # needs sage.rings.finite_rings
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                                               # needs sage.rings.finite_rings
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)                                             # needs sage.rings.finite_rings
True

and pushout works:

sage: # needs sage.rings.finite_rings
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: # needs sage.rings.finite_rings
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: # needs sage.rings.finite_rings
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: # needs sage.rings.finite_rings
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: Section

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