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

>>> from sage.all import *
>>> R = QQ['t']; (t,) = R._first_ngens(1)
>>> der = R.derivation()
>>> A = R['d', der]; (d,) = A._first_ngens(1)
>>> 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

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
[Python] 
>>> from sage.all import *
>>> f = Integer(1)/d
>>> f
d^(-1)
>>> 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

>>> from sage.all import *
>>> g = t / d
>>> 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
[Python] 
>>> from sage.all import *
>>> g.left_numerator()
t
>>> 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

>>> from sage.all import *
>>> g.left_denominator()
d - 1/t
>>> 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: 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
[Python] 
>>> from sage.all import *
>>> sigma = R.hom([t**Integer(2)])
>>> S = R['x', sigma]; (x,) = S._first_ngens(1)
>>> F = S.fraction_field()
>>> f = F.random_element()
>>> while not f:
...     f = F.random_element()
>>> 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

>>> from sage.all import *
>>> P = d**Integer(2) + t*d + Integer(1)
>>> Q = d + t**Integer(2)
>>> D = d**Integer(3) + t**Integer(2) + Integer(1)
>>> f = P**(-Integer(1)) * Q
>>> f
(d^2 + t*d + 1)^(-1) * (d + t^2)
>>> g = (D*P)**(-Integer(1)) * (D*Q)
>>> g
(d^2 + t*d + 1)^(-1) * (d + t^2)

>>> P = x**Integer(2) + t*x + Integer(1)
>>> Q = x + t**Integer(2)
>>> D = x**Integer(3) + t**Integer(2) + Integer(1)
>>> f = P**(-Integer(1)) * Q
>>> f
(x^2 + t*x + 1)^(-1) * (x + t^2)
>>> g = (D*P)**(-Integer(1)) * (D*Q)
>>> 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)
>>> 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)
[Python] 
>>> from sage.all import *
>>> f = Integer(1) / d
>>> g = Integer(1) / (d + t)
>>> u = f + g; u
(d^2 + ((t^2 - 1)/t)*d)^(-1) * (2*d + (t^2 - 2)/t)
>>> v = f - g; v
(d^2 + ((t^2 - 1)/t)*d)^(-1) * t
>>> u + v
d^(-1) * 2

>>> f * g
(d^2 + t*d)^(-1)
>>> 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)

>>> from sage.all import *
>>> g * f
(d^2 + t*d + 1)^(-1)
>>> g**(-Integer(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)[source]

Bases: RingHomomorphism

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

section()[source]

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)

>>> from sage.all import *
>>> k = GF(Integer(5)**Integer(3), names=('a',)); (a,) = k._first_ngens(1)
>>> S = SkewPolynomialRing(k, k.frobenius_endomorphism(), names=('x',)); (x,) = S._first_ngens(1)
>>> K = S.fraction_field()
>>> Z = K.center()
>>> iota = K.coerce_map_from(Z)
>>> sigma = iota.section()
>>> sigma(x**Integer(3) / (x**Integer(6) + Integer(1)))
z/(z^2 + 1)
class sage.rings.polynomial.ore_function_field.OreFunctionField(ring, category=None)[source]

Bases: Parent, UniqueRepresentation

A class for fraction fields of Ore polynomial rings.

Element = None
change_var(var)[source]

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: 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

>>> from sage.all import *
>>> k = GF(Integer(5)**Integer(3), names=('t',)); (t,) = k._first_ngens(1)
>>> Frob = k.frobenius_endomorphism()
>>> R = OrePolynomialRing(k,Frob, names=('x',)); (x,) = R._first_ngens(1)
>>> K = R.fraction_field()
>>> K
Ore Function Field in x over Finite Field in t of size 5^3 twisted by t |--> t^5
>>> 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
>>> Ky is K.change_var('y')
True
characteristic()[source]

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

>>> from sage.all import *
>>> R = QQ['t']; (t,) = R._first_ngens(1)
>>> sigma = R.hom([t+Integer(1)])
>>> S = R['x',sigma]
>>> S.fraction_field().characteristic()
0

>>> k = GF(Integer(5)**Integer(3), names=('u',)); (u,) = k._first_ngens(1)
>>> Frob = k.frobenius_endomorphism()
>>> S = k['y',Frob]
>>> S.fraction_field().characteristic()
5
fraction_field()[source]

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

>>> from sage.all import *
>>> R = QQ['t']; (t,) = R._first_ngens(1)
>>> der = R.derivation()
>>> A = R['d', der]; (d,) = A._first_ngens(1)
>>> 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
>>> K.fraction_field()
Ore Function Field in d
 over Fraction Field of Univariate Polynomial Ring in t over Rational Field
 twisted by d/dt
>>> K.fraction_field() is K
True
gen(n=0)[source]

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

>>> from sage.all import *
>>> k = GF(Integer(5)**Integer(4), names=('a',)); (a,) = k._first_ngens(1)
>>> Frob = k.frobenius_endomorphism()
>>> S = k['x', Frob]; (x,) = S._first_ngens(1)
>>> K = S.fraction_field()
>>> K.gen()
x
gens()[source]

Return the tuple of generators of self.

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.gens()
(x,)

>>> from sage.all import *
>>> k = GF(Integer(5)**Integer(4), names=('a',)); (a,) = k._first_ngens(1)
>>> Frob = k.frobenius_endomorphism()
>>> S = k['x', Frob]; (x,) = S._first_ngens(1)
>>> K = S.fraction_field()
>>> K.gens()
(x,)
gens_dict()[source]

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}

>>> from sage.all import *
>>> R = ZZ['t']; (t,) = R._first_ngens(1)
>>> sigma = R.hom([t+Integer(1)])
>>> S = OrePolynomialRing(R, sigma, names=('x',)); (x,) = S._first_ngens(1)
>>> K = S.fraction_field()
>>> K.gens_dict()
{'x': x}
is_exact()[source]

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

>>> from sage.all import *
>>> k = GF(Integer(5)**Integer(3), names=('t',)); (t,) = k._first_ngens(1)
>>> Frob = k.frobenius_endomorphism()
>>> S = k['x', Frob]; (x,) = S._first_ngens(1)
>>> K = S.fraction_field()
>>> K.is_exact()
True

>>> k = Qq(Integer(5)**Integer(3), names=('u',)); (u,) = k._first_ngens(1)
>>> Frob = k.frobenius_endomorphism()
>>> S = k['x', Frob]; (x,) = S._first_ngens(1)
>>> K = S.fraction_field()
>>> K.is_exact()
False
is_field(proof=False)[source]

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

>>> from sage.all import *
>>> k = GF(Integer(5)**Integer(3), names=('a',)); (a,) = k._first_ngens(1)
>>> Frob = k.frobenius_endomorphism()
>>> S = k['x', Frob]; (x,) = S._first_ngens(1)
>>> K = S.fraction_field()
>>> S.is_field()
False
>>> K.is_field()
True
is_finite()[source]

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

>>> from sage.all import *
>>> k = GF(Integer(5)**Integer(3), names=('t',)); (t,) = k._first_ngens(1)
>>> k.is_finite()
True
>>> Frob = k.frobenius_endomorphism()
>>> S = k['x',Frob]; (x,) = S._first_ngens(1)
>>> K = S.fraction_field()
>>> K.is_finite()
False
is_sparse()[source]

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

>>> from sage.all import *
>>> R = RR['t']; (t,) = R._first_ngens(1)
>>> sigma = R.hom([t+Integer(1)])
>>> S = R['x', sigma]; (x,) = S._first_ngens(1)
>>> K = S.fraction_field()
>>> K.is_sparse()
False
ngens()[source]

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

>>> from sage.all import *
>>> R = RR['t']; (t,) = R._first_ngens(1)
>>> sigma = R.hom([t+Integer(1)])
>>> S = R['x',sigma]; (x,) = S._first_ngens(1)
>>> K = S.fraction_field()
>>> K.ngens()
1
parameter(n=0)[source]

alias of gen().

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

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: 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)

>>> from sage.all import *
>>> k = GF(Integer(5)**Integer(3), names=('t',)); (t,) = k._first_ngens(1)
>>> Frob = k.frobenius_endomorphism()
>>> S = k['x', Frob]; (x,) = S._first_ngens(1)
>>> K = S.fraction_field()
>>> 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)
>>> 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)
>>> K.random_element(degree=Integer(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)
>>> K.random_element(degree=[Integer(2),Integer(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()[source]

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()

>>> from sage.all import *
>>> R = QQ['t']; (t,) = R._first_ngens(1)
>>> der = R.derivation(); der
d/dt
>>> A = R['d', der]; (d,) = A._first_ngens(1)
>>> F = A.fraction_field()
>>> F.twisting_derivation()
d/dt

>>> k = GF(Integer(5)**Integer(3), names=('a',)); (a,) = k._first_ngens(1)
>>> Frob = k.frobenius_endomorphism()
>>> S = k['x', Frob]; (x,) = S._first_ngens(1)
>>> K = S.fraction_field()
>>> K.twisting_derivation()

See also

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

twisting_morphism(n=1)[source]

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()
sage: K.twisting_morphism()
Ring endomorphism of
 Fraction Field of Univariate Polynomial Ring in t over Rational Field
  Defn: t |--> t + 1

>>> from sage.all import *
>>> R = QQ['t']; (t,) = R._first_ngens(1)
>>> sigma = R.hom([t+Integer(1)])
>>> S = R['x', sigma]; (x,) = S._first_ngens(1)
>>> K = S.fraction_field()
>>> 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()
[Python] 
>>> from sage.all import *
>>> der = R.derivation()
>>> A = R['x', der]; (d,) = A._first_ngens(1)
>>> F = A.fraction_field()
>>> 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)[source]

Bases: OreFunctionField

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

center(name=None, names=None, default=False)[source]

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: 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

>>> from sage.all import *
>>> k = GF(Integer(5)**Integer(3), names=('t',)); (t,) = k._first_ngens(1)
>>> Frob = k.frobenius_endomorphism()
>>> S = k['x',Frob]; (x,) = S._first_ngens(1)
>>> K = S.fraction_field()
>>> 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
[Python] 
>>> from sage.all import *
>>> 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

>>> from sage.all import *
>>> Zy = K.center(names=('y',)); (y,) = Zy._first_ngens(1); 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
[Python] 
>>> from sage.all import *
>>> 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

>>> from sage.all import *
>>> x.parent()
Ore Polynomial Ring in x over Finite Field in t of size 5^3 twisted by t |--> t^5
>>> y.parent()
Fraction Field of Univariate Polynomial Ring in y over Finite Field of size 5
>>> P = x + y; P
x^3 + x
>>> 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
[Python] 
>>> from sage.all import *
>>> Zy(x**(-Integer(6)) + Integer(2))
(2*y^2 + 1)/y^2
>>> Zy(Integer(1)/x**Integer(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

>>> from sage.all import *
>>> k = GF(Integer(11)**Integer(3), names=('t',)); (t,) = k._first_ngens(1)
>>> phi = k.frobenius_endomorphism()
>>> S = k['X', phi]; (X,) = S._first_ngens(1)
>>> K = S.fraction_field()
>>> C = K.center(names=('u',)); (u,) = C._first_ngens(1)# first call
>>> C
Fraction Field of Univariate Polynomial Ring in u over Finite Field of size 11
>>> 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
[Python] 
>>> from sage.all import *
>>> D = K.center(default=True, names=('v',)); (v,) = D._first_ngens(1)
>>> D
Fraction Field of Univariate Polynomial Ring in v over Finite Field of size 11
>>> 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)[source]

Bases: Section

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