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

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

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

f = 1/d
f
f.parent() is K

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

g = t / d
g

The left numerator and right denominator are accessible as follows:

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

g.left_numerator()
g.right_denominator()

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

g.left_denominator()
g.right_numerator()

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

sigma = R.hom([t^2])
S.<x> = R['x', sigma]
F = S.fraction_field()
f = F.random_element()
while not f:
    f = F.random_element()
f.left_numerator()

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

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

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)

f = 1 / d
g = 1 / (d + t)
u = f + g; u
v = f - g; v
u + v
f * g
f / g

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)

g * f
g^(-1) * f

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: 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)

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

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

Bases: Algebra, 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 – 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

k.<t> = GF(5^3)
Frob = k.frobenius_endomorphism()
R.<x> = OrePolynomialRing(k,Frob)
K = R.fraction_field()
K
Ky = K.change_var('y'); Ky
Ky is K.change_var('y')

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

R.<t> = QQ[]
sigma = R.hom([t+1])
S = R['x',sigma]
S.fraction_field().characteristic()
k.<u> = GF(5^3)
Frob = k.frobenius_endomorphism()
S = k['y',Frob]
S.fraction_field().characteristic()

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

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

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

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

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}

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

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

k.<a> = GF(5^3)
Frob = k.frobenius_endomorphism()
S.<x> = k['x', Frob]
K = S.fraction_field()
K.is_commutative()
T.<y> = k['y', Frob^3]
L = T.fraction_field()
L.is_commutative()

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

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

is_field(proof=False)#

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

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

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

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

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

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

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

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

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

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

k.<t> = GF(5^3)
Frob = k.frobenius_endomorphism()
S.<x> = k['x', Frob]
K = S.fraction_field()
K.random_element()              # random
K.random_element(monic=True)    # random
K.random_element(degree=3)      # random
K.random_element(degree=[2,5])  # random

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()

R.<t> = QQ[]
der = R.derivation(); der
A.<d> = R['d', der]
F = A.fraction_field()
F.twisting_derivation()
k.<a> = GF(5^3)
Frob = k.frobenius_endomorphism()
S.<x> = k['x', Frob]
K = S.fraction_field()
K.twisting_derivation()