Univariate Ore polynomial rings¶

This module provides the OrePolynomialRing, which constructs a general dense univariate Ore polynomial ring over a commutative base with equipped with an endomorphism and/or a derivation.

AUTHOR:

Xavier Caruso (2020-04)

class sage.rings.polynomial.ore_polynomial_ring.OrePolynomialRing(base_ring, morphism, derivation, name, sparse, category=None)[source]¶

Bases: UniqueRepresentation, Parent

Construct and return the globally unique Ore polynomial ring with the given properties and variable names.

Given a ring \(R\) and a ring automorphism \(\sigma\) of \(R\) and a \(\sigma\)-derivation \(\partial\), the ring of Ore polynomials \(R[X; \sigma, \partial]\) is the usual abelian group polynomial \(R[X]\) equipped with the modification multiplication deduced from the rule \(X a = \sigma(a) X + \partial(a)\). We refer to [Ore1933] for more material on Ore polynomials.

INPUT:

base_ring – a commutative ring
twisting_map – either an endomorphism of the base ring, or a (twisted) derivation of it
names – string or list of strings
sparse – boolean (default: False); currently not supported

EXAMPLES:

The case of a twisting endomorphism

We create the Ore ring \(\GF{5^3}[x, \text{Frob}]\) where Frob is the Frobenius endomorphism:

Sage

sage: # needs sage.rings.finite_rings
sage: k.<a> = GF(5^3)
sage: Frob = k.frobenius_endomorphism()
sage: S = OrePolynomialRing(k, Frob, 'x'); S
Ore Polynomial Ring in x over Finite Field in a of size 5^3 twisted by a |--> a^5

Python

>>> from sage.all import *
>>> # needs sage.rings.finite_rings
>>> k = GF(Integer(5)**Integer(3), names=('a',)); (a,) = k._first_ngens(1)
>>> Frob = k.frobenius_endomorphism()
>>> S = OrePolynomialRing(k, Frob, 'x'); S
Ore Polynomial Ring in x over Finite Field in a of size 5^3 twisted by a |--> a^5

Sage Live

# needs sage.rings.finite_rings
k.<a> = GF(5^3)
Frob = k.frobenius_endomorphism()
S = OrePolynomialRing(k, Frob, 'x'); S

In particular, observe that it is not needed to create and pass in the twisting derivation (which is \(0\) in our example).

As a shortcut, we can use the square brackets notation as follow:

Sage

sage: # needs sage.rings.finite_rings
sage: T.<x> = k['x', Frob]; T
Ore Polynomial Ring in x over Finite Field in a of size 5^3 twisted by a |--> a^5
sage: T is S
True

Python

>>> from sage.all import *
>>> # needs sage.rings.finite_rings
>>> T = k['x', Frob]; (x,) = T._first_ngens(1); T
Ore Polynomial Ring in x over Finite Field in a of size 5^3 twisted by a |--> a^5
>>> T is S
True

Sage Live

# needs sage.rings.finite_rings
T.<x> = k['x', Frob]; T
T is S

We emphasize that it is necessary to repeat the name of the variable in the right hand side. Indeed, the following fails (it is interpreted by Sage as a classical polynomial ring with variable name Frob):

Sage

sage: T.<x> = k[Frob]                                                           # needs sage.rings.finite_rings
Traceback (most recent call last):
...
ValueError: variable name 'Frobenius endomorphism a |--> a^5 on
Finite Field in a of size 5^3' is not alphanumeric

Python

>>> from sage.all import *
>>> T = k[Frob]; (x,) = T._first_ngens(1)# needs sage.rings.finite_rings
Traceback (most recent call last):
...
ValueError: variable name 'Frobenius endomorphism a |--> a^5 on
Finite Field in a of size 5^3' is not alphanumeric

Sage Live

T.<x> = k[Frob]                                                           # needs sage.rings.finite_rings

Note moreover that, similarly to the classical case, using the brackets notation also sets the variable:

Sage

sage: x.parent() is S                                                           # needs sage.rings.finite_rings
True

Python

>>> from sage.all import *
>>> x.parent() is S                                                           # needs sage.rings.finite_rings
True

Sage Live

x.parent() is S                                                           # needs sage.rings.finite_rings

We are now ready to carry on computations in the Ore ring:

Sage

sage: x*a                                                                       # needs sage.rings.finite_rings
(2*a^2 + 4*a + 4)*x
sage: Frob(a)*x                                                                 # needs sage.rings.finite_rings
(2*a^2 + 4*a + 4)*x

Python

>>> from sage.all import *
>>> x*a                                                                       # needs sage.rings.finite_rings
(2*a^2 + 4*a + 4)*x
>>> Frob(a)*x                                                                 # needs sage.rings.finite_rings
(2*a^2 + 4*a + 4)*x

Sage Live

x*a                                                                       # needs sage.rings.finite_rings
Frob(a)*x                                                                 # needs sage.rings.finite_rings

The case of a twisting derivation

We can similarly create the Ore ring of differential operators over \(\QQ[t]\), namely \(\QQ[t][d, \frac{d}{dt}]\):

Sage

sage: # needs sage.rings.finite_rings
sage: R.<t> = QQ[]
sage: der = R.derivation(); der
d/dt
sage: A = OrePolynomialRing(R, der, 'd'); A
Ore Polynomial Ring in d over Univariate Polynomial Ring in t
 over Rational Field twisted by d/dt

Python

>>> from sage.all import *
>>> # needs sage.rings.finite_rings
>>> R = QQ['t']; (t,) = R._first_ngens(1)
>>> der = R.derivation(); der
d/dt
>>> A = OrePolynomialRing(R, der, 'd'); A
Ore Polynomial Ring in d over Univariate Polynomial Ring in t
 over Rational Field twisted by d/dt

Sage Live

# needs sage.rings.finite_rings
R.<t> = QQ[]
der = R.derivation(); der
A = OrePolynomialRing(R, der, 'd'); A

Again, the brackets notation is available:

Sage

sage: B.<d> = R['d', der]                                                       # needs sage.rings.finite_rings
sage: A is B                                                                    # needs sage.rings.finite_rings
True

Python

>>> from sage.all import *
>>> B = R['d', der]; (d,) = B._first_ngens(1)# needs sage.rings.finite_rings
>>> A is B                                                                    # needs sage.rings.finite_rings
True

Sage Live

B.<d> = R['d', der]                                                       # needs sage.rings.finite_rings
A is B                                                                    # needs sage.rings.finite_rings

and computations can be carried out:

Sage

sage: d*t                                                                       # needs sage.rings.finite_rings
t*d + 1

Python

>>> from sage.all import *
>>> d*t                                                                       # needs sage.rings.finite_rings
t*d + 1

Sage Live

d*t                                                                       # needs sage.rings.finite_rings

The combined case

Ore polynomial rings involving at the same time a twisting morphism \(\sigma\) and a twisting \(\sigma\)-derivation can be created as well as follows:

Sage

sage: # needs sage.rings.padics
sage: F.<u> = Qq(3^2)
sage: sigma = F.frobenius_endomorphism(); sigma
Frobenius endomorphism on 3-adic Unramified Extension Field in u
 defined by x^2 + 2*x + 2 lifting u |--> u^3 on the residue field
sage: der = F.derivation(3, twist=sigma); der
(3 + O(3^21))*([Frob] - id)
sage: M.<X> = F['X', der]; M
Ore Polynomial Ring in X over 3-adic Unramified Extension Field in u
 defined by x^2 + 2*x + 2 twisted by Frob and (3 + O(3^21))*([Frob] - id)

Python

>>> from sage.all import *
>>> # needs sage.rings.padics
>>> F = Qq(Integer(3)**Integer(2), names=('u',)); (u,) = F._first_ngens(1)
>>> sigma = F.frobenius_endomorphism(); sigma
Frobenius endomorphism on 3-adic Unramified Extension Field in u
 defined by x^2 + 2*x + 2 lifting u |--> u^3 on the residue field
>>> der = F.derivation(Integer(3), twist=sigma); der
(3 + O(3^21))*([Frob] - id)
>>> M = F['X', der]; (X,) = M._first_ngens(1); M
Ore Polynomial Ring in X over 3-adic Unramified Extension Field in u
 defined by x^2 + 2*x + 2 twisted by Frob and (3 + O(3^21))*([Frob] - id)

Sage Live

# needs sage.rings.padics
F.<u> = Qq(3^2)
sigma = F.frobenius_endomorphism(); sigma
der = F.derivation(3, twist=sigma); der
M.<X> = F['X', der]; M

We emphasize that we only need to pass in the twisted derivation as it already contains in it the datum of the twisting endomorphism. Actually, passing in both twisting maps results in an error:

Sage

sage: F['X', sigma, der]                                                        # needs sage.rings.padics
Traceback (most recent call last):
...
ValueError: variable name 'Frobenius endomorphism ...' is not alphanumeric

Python

>>> from sage.all import *
>>> F['X', sigma, der]                                                        # needs sage.rings.padics
Traceback (most recent call last):
...
ValueError: variable name 'Frobenius endomorphism ...' is not alphanumeric

Sage Live

F['X', sigma, der]                                                        # needs sage.rings.padics

Examples of variable name context

Consider the following:

Sage

sage: R.<t> = ZZ[]
sage: sigma = R.hom([t+1])
sage: S.<x> = SkewPolynomialRing(R, sigma); S
Ore Polynomial Ring in x over Univariate Polynomial Ring in t over Integer Ring
 twisted by t |--> t + 1

Python

>>> from sage.all import *
>>> R = ZZ['t']; (t,) = R._first_ngens(1)
>>> sigma = R.hom([t+Integer(1)])
>>> S = SkewPolynomialRing(R, sigma, names=('x',)); (x,) = S._first_ngens(1); S
Ore Polynomial Ring in x over Univariate Polynomial Ring in t over Integer Ring
 twisted by t |--> t + 1

Sage Live

R.<t> = ZZ[]
sigma = R.hom([t+1])
S.<x> = SkewPolynomialRing(R, sigma); S

The names of the variables defined above cannot be arbitrarily modified because each Ore polynomial ring is unique in Sage and other objects in Sage could have pointers to that Ore polynomial ring.

However, the variable can be changed within the scope of a with block using the localvars context:

Sage

sage: R.<t> = ZZ[]
sage: sigma = R.hom([t+1])
sage: S.<x> = SkewPolynomialRing(R, sigma); S
Ore Polynomial Ring in x over Univariate Polynomial Ring in t over Integer Ring
 twisted by t |--> t + 1

sage: with localvars(S, ['y']):
....:     print(S)
Ore Polynomial Ring in y over Univariate Polynomial Ring in t over Integer Ring
 twisted by t |--> t + 1

Python

>>> from sage.all import *
>>> R = ZZ['t']; (t,) = R._first_ngens(1)
>>> sigma = R.hom([t+Integer(1)])
>>> S = SkewPolynomialRing(R, sigma, names=('x',)); (x,) = S._first_ngens(1); S
Ore Polynomial Ring in x over Univariate Polynomial Ring in t over Integer Ring
 twisted by t |--> t + 1

>>> with localvars(S, ['y']):
...     print(S)
Ore Polynomial Ring in y over Univariate Polynomial Ring in t over Integer Ring
 twisted by t |--> t + 1

Sage Live

R.<t> = ZZ[]
sigma = R.hom([t+1])
S.<x> = SkewPolynomialRing(R, sigma); S
with localvars(S, ['y']):
    print(S)

Uniqueness and immutability

In Sage, there is exactly one Ore polynomial ring for each quadruple (base ring, twisting morphism, twisting derivation, name of the variable):

Sage

sage: # needs sage.rings.finite_rings
sage: k.<a> = GF(7^3)
sage: Frob = k.frobenius_endomorphism()
sage: S = k['x', Frob]
sage: T = k['x', Frob]
sage: S is T
True

Python

>>> from sage.all import *
>>> # needs sage.rings.finite_rings
>>> k = GF(Integer(7)**Integer(3), names=('a',)); (a,) = k._first_ngens(1)
>>> Frob = k.frobenius_endomorphism()
>>> S = k['x', Frob]
>>> T = k['x', Frob]
>>> S is T
True

Sage Live

# needs sage.rings.finite_rings
k.<a> = GF(7^3)
Frob = k.frobenius_endomorphism()
S = k['x', Frob]
T = k['x', Frob]
S is T

Rings with different variables names are different:

Sage

sage: S is k['y', Frob]                                                         # needs sage.rings.finite_rings
False

Python

>>> from sage.all import *
>>> S is k['y', Frob]                                                         # needs sage.rings.finite_rings
False

Sage Live

S is k['y', Frob]                                                         # needs sage.rings.finite_rings

Similarly, varying the twisting morphisms yields to different Ore rings (expect when the morphism coincide):

Sage

sage: S is k['x', Frob^2]                                                       # needs sage.rings.finite_rings
False
sage: S is k['x', Frob^3]                                                       # needs sage.rings.finite_rings
False
sage: S is k['x', Frob^4]                                                       # needs sage.rings.finite_rings
True

Python

>>> from sage.all import *
>>> S is k['x', Frob**Integer(2)]                                                       # needs sage.rings.finite_rings
False
>>> S is k['x', Frob**Integer(3)]                                                       # needs sage.rings.finite_rings
False
>>> S is k['x', Frob**Integer(4)]                                                       # needs sage.rings.finite_rings
True

Sage Live

S is k['x', Frob^2]                                                       # needs sage.rings.finite_rings
S is k['x', Frob^3]                                                       # needs sage.rings.finite_rings
S is k['x', Frob^4]                                                       # needs sage.rings.finite_rings

Todo

Sparse Ore Polynomial Ring
Multivariate Ore Polynomial Ring

Element = None¶

change_var(var)[source]¶

Return the Ore polynomial ring 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

sage: # needs sage.rings.finite_rings
sage: k.<t> = GF(5^3)
sage: Frob = k.frobenius_endomorphism()
sage: R.<x> = OrePolynomialRing(k,Frob); R
Ore Polynomial Ring in x over Finite Field in t of size 5^3 twisted by t |--> t^5
sage: Ry = R.change_var('y'); Ry
Ore Polynomial Ring in y over Finite Field in t of size 5^3 twisted by t |--> t^5
sage: Ry is R.change_var('y')
True

Python

>>> from sage.all import *
>>> # needs sage.rings.finite_rings
>>> 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); R
Ore Polynomial Ring in x over Finite Field in t of size 5^3 twisted by t |--> t^5
>>> Ry = R.change_var('y'); Ry
Ore Polynomial Ring in y over Finite Field in t of size 5^3 twisted by t |--> t^5
>>> Ry is R.change_var('y')
True

Sage Live

# needs sage.rings.finite_rings
k.<t> = GF(5^3)
Frob = k.frobenius_endomorphism()
R.<x> = OrePolynomialRing(k,Frob); R
Ry = R.change_var('y'); Ry
Ry is R.change_var('y')

characteristic()[source]¶

Return the characteristic of the base ring of self.

EXAMPLES:

Sage

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

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

Python

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

>>> # needs sage.rings.finite_rings
>>> k = GF(Integer(5)**Integer(3), names=('u',)); (u,) = k._first_ngens(1)
>>> Frob = k.frobenius_endomorphism()
>>> k['y',Frob].characteristic()
5

Sage Live

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

fraction_field()[source]¶

Return the fraction field of this skew ring.

EXAMPLES:

Sage

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(); K
Ore Function Field in x over Finite Field in a of size 5^3 twisted by a |--> a^5
sage: f = 1/(x + a); f
(x + a)^(-1)
sage: f.parent() is K
True

Python

>>> from sage.all import *
>>> # needs sage.rings.finite_rings
>>> 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
Ore Function Field in x over Finite Field in a of size 5^3 twisted by a |--> a^5
>>> f = Integer(1)/(x + a); f
(x + a)^(-1)
>>> f.parent() is K
True

Sage Live

# needs sage.rings.finite_rings
k.<a> = GF(5^3)
Frob = k.frobenius_endomorphism()
S.<x> = k['x', Frob]
K = S.fraction_field(); K
f = 1/(x + a); f
f.parent() is K

Below is another example with differentiel operators:

Sage

sage: R.<t> = QQ[]
sage: der = R.derivation()
sage: A.<d> = R['d', der]
sage: A.fraction_field()
Ore Function Field in d over Fraction Field of Univariate Polynomial Ring in t
 over Rational Field twisted by d/dt
sage: f = t/d; f
(d - 1/t)^(-1) * t
sage: f*d
t

Python

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

Sage Live

R.<t> = QQ[]
der = R.derivation()
A.<d> = R['d', der]
A.fraction_field()
f = t/d; f
f*d

gen(n=0)[source]¶

Return the indeterminate generator of this Ore polynomial ring.

INPUT:

n – index of generator to return (default: 0); exists for compatibility with other polynomial rings

EXAMPLES:

Sage

sage: R.<t> = QQ[]
sage: sigma = R.hom([t+1])
sage: S.<x> = R['x',sigma]; S
Ore Polynomial Ring in x over Univariate Polynomial Ring in t
 over Rational Field twisted by t |--> t + 1
sage: y = S.gen(); y
x
sage: y == x
True
sage: S.gen(0)
x

Python

>>> 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); S
Ore Polynomial Ring in x over Univariate Polynomial Ring in t
 over Rational Field twisted by t |--> t + 1
>>> y = S.gen(); y
x
>>> y == x
True
>>> S.gen(Integer(0))
x

Sage Live

R.<t> = QQ[]
sigma = R.hom([t+1])
S.<x> = R['x',sigma]; S
y = S.gen(); y
y == x
S.gen(0)

This is also known as the parameter:

Sage

sage: S.parameter() is S.gen()
True

Python

>>> from sage.all import *
>>> S.parameter() is S.gen()
True

Sage Live

S.parameter() is S.gen()

gens()[source]¶

Return the tuple of generators of self.

EXAMPLES:

Sage

sage: R.<t> = QQ[]
sage: sigma = R.hom([t+1])
sage: S.<x> = R['x',sigma]; S
Ore Polynomial Ring in x over Univariate Polynomial Ring in t
 over Rational Field twisted by t |--> t + 1
sage: S.gens()
(x,)

Python

>>> 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); S
Ore Polynomial Ring in x over Univariate Polynomial Ring in t
 over Rational Field twisted by t |--> t + 1
>>> S.gens()
(x,)

Sage Live

R.<t> = QQ[]
sigma = R.hom([t+1])
S.<x> = R['x',sigma]; S
S.gens()

gens_dict()[source]¶

Return a {name: variable} dictionary of the generators of this Ore polynomial ring.

EXAMPLES:

Sage

sage: R.<t> = ZZ[]
sage: sigma = R.hom([t+1])
sage: S.<x> = SkewPolynomialRing(R,sigma)
sage: S.gens_dict()
{'x': x}

Python

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

Sage Live

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

is_commutative()[source]¶

Return True if this Ore polynomial ring is commutative.

This holds if the twisting morphism is the identity and the twisting derivation vanishes.

EXAMPLES:

Sage

sage: # needs sage.rings.finite_rings
sage: k.<a> = GF(5^3)
sage: Frob = k.frobenius_endomorphism()
sage: S.<x> = k['x', Frob]
sage: S.is_commutative()
False
sage: T.<y> = k['y', Frob^3]
sage: T.is_commutative()
True

sage: R.<t> = GF(5)[]
sage: der = R.derivation()
sage: A.<d> = R['d', der]
sage: A.is_commutative()
False
sage: B.<b> = R['b', 5*der]
sage: B.is_commutative()
True

Python

>>> from sage.all import *
>>> # needs sage.rings.finite_rings
>>> 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)
>>> S.is_commutative()
False
>>> T = k['y', Frob**Integer(3)]; (y,) = T._first_ngens(1)
>>> T.is_commutative()
True

>>> R = GF(Integer(5))['t']; (t,) = R._first_ngens(1)
>>> der = R.derivation()
>>> A = R['d', der]; (d,) = A._first_ngens(1)
>>> A.is_commutative()
False
>>> B = R['b', Integer(5)*der]; (b,) = B._first_ngens(1)
>>> B.is_commutative()
True

Sage Live

# needs sage.rings.finite_rings
k.<a> = GF(5^3)
Frob = k.frobenius_endomorphism()
S.<x> = k['x', Frob]
S.is_commutative()
T.<y> = k['y', Frob^3]
T.is_commutative()
R.<t> = GF(5)[]
der = R.derivation()
A.<d> = R['d', der]
A.is_commutative()
B.<b> = R['b', 5*der]
B.is_commutative()

is_exact()[source]¶

Return True if elements of this Ore polynomial ring are exact.

This happens if and only if elements of the base ring are exact.

EXAMPLES:

Sage

sage: # needs sage.rings.finite_rings
sage: k.<t> = GF(5^3)
sage: Frob = k.frobenius_endomorphism()
sage: S.<x> = k['x', Frob]
sage: S.is_exact()
True
sage: S.base_ring().is_exact()
True
sage: R.<u> = k[[]]
sage: sigma = R.hom([u + u^2])
sage: T.<y> = R['y', sigma]
sage: T.is_exact()
False
sage: T.base_ring().is_exact()
False

Python

>>> from sage.all import *
>>> # needs sage.rings.finite_rings
>>> 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)
>>> S.is_exact()
True
>>> S.base_ring().is_exact()
True
>>> R = k[['u']]; (u,) = R._first_ngens(1)
>>> sigma = R.hom([u + u**Integer(2)])
>>> T = R['y', sigma]; (y,) = T._first_ngens(1)
>>> T.is_exact()
False
>>> T.base_ring().is_exact()
False

Sage Live

# needs sage.rings.finite_rings
k.<t> = GF(5^3)
Frob = k.frobenius_endomorphism()
S.<x> = k['x', Frob]
S.is_exact()
S.base_ring().is_exact()
R.<u> = k[[]]
sigma = R.hom([u + u^2])
T.<y> = R['y', sigma]
T.is_exact()
T.base_ring().is_exact()

is_field(proof=False)[source]¶

Return always False since Ore polynomial rings are never fields.

EXAMPLES:

Sage

sage: # needs sage.rings.finite_rings
sage: k.<a> = GF(5^3)
sage: Frob = k.frobenius_endomorphism()
sage: S.<x> = k['x', Frob]
sage: S.is_field()
False

Python

>>> from sage.all import *
>>> # needs sage.rings.finite_rings
>>> 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)
>>> S.is_field()
False

Sage Live

# needs sage.rings.finite_rings
k.<a> = GF(5^3)
Frob = k.frobenius_endomorphism()
S.<x> = k['x', Frob]
S.is_field()

is_finite()[source]¶

Return False since Ore polynomial rings are not finite (unless the base ring is \(0\)).

EXAMPLES:

Sage

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: S.is_finite()
False

Python

>>> from sage.all import *
>>> # needs sage.rings.finite_rings
>>> 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)
>>> S.is_finite()
False

Sage Live

# needs sage.rings.finite_rings
k.<t> = GF(5^3)
k.is_finite()
Frob = k.frobenius_endomorphism()
S.<x> = k['x',Frob]
S.is_finite()

is_sparse()[source]¶

Return True if the elements of this Ore polynomial ring are sparsely represented.

Warning

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

EXAMPLES:

Sage

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

Python

>>> 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)
>>> S.is_sparse()
False

Sage Live

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

ngens()[source]¶

Return the number of generators of this Ore polynomial ring.

This is \(1\).

EXAMPLES:

Sage

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

Python

>>> 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)
>>> S.ngens()
1

Sage Live

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

parameter(n=0)[source]¶

Return the indeterminate generator of this Ore polynomial ring.

INPUT:

n – index of generator to return (default: 0); exists for compatibility with other polynomial rings

EXAMPLES:

Sage

sage: R.<t> = QQ[]
sage: sigma = R.hom([t+1])
sage: S.<x> = R['x',sigma]; S
Ore Polynomial Ring in x over Univariate Polynomial Ring in t
 over Rational Field twisted by t |--> t + 1
sage: y = S.gen(); y
x
sage: y == x
True
sage: S.gen(0)
x

Python

>>> 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); S
Ore Polynomial Ring in x over Univariate Polynomial Ring in t
 over Rational Field twisted by t |--> t + 1
>>> y = S.gen(); y
x
>>> y == x
True
>>> S.gen(Integer(0))
x

Sage Live

R.<t> = QQ[]
sigma = R.hom([t+1])
S.<x> = R['x',sigma]; S
y = S.gen(); y
y == x
S.gen(0)

This is also known as the parameter:

Sage

sage: S.parameter() is S.gen()
True

Python

>>> from sage.all import *
>>> S.parameter() is S.gen()
True

Sage Live

S.parameter() is S.gen()

random_element(degree=(-1, 2), monic=False, *args, **kwds)[source]¶

Return a random Ore polynomial in this ring.

INPUT:

degree – (default: (-1,2)) integer with degree or a tuple of integers with minimum and maximum degrees
monic – boolean (default: False); if True, return a monic Ore polynomial
*args, **kwds – passed on to the random_element method for the base ring

OUTPUT:

Ore polynomial such that the coefficients of \(x^i\), for \(i\) up to degree, are random elements from the base ring, randomized subject to the arguments *args and **kwds.

EXAMPLES:

Sage

sage: # needs sage.rings.finite_rings
sage: k.<t> = GF(5^3)
sage: Frob = k.frobenius_endomorphism()
sage: S.<x> = k['x', Frob]
sage: S.random_element()  # random
(2*t^2 + 3)*x^2 + (4*t^2 + t + 4)*x + 2*t^2 + 2
sage: S.random_element(monic=True)  # random
x^2 + (2*t^2 + t + 1)*x + 3*t^2 + 3*t + 2

Python

>>> from sage.all import *
>>> # needs sage.rings.finite_rings
>>> 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)
>>> S.random_element()  # random
(2*t^2 + 3)*x^2 + (4*t^2 + t + 4)*x + 2*t^2 + 2
>>> S.random_element(monic=True)  # random
x^2 + (2*t^2 + t + 1)*x + 3*t^2 + 3*t + 2

Sage Live

# needs sage.rings.finite_rings
k.<t> = GF(5^3)
Frob = k.frobenius_endomorphism()
S.<x> = k['x', Frob]
S.random_element()  # random
S.random_element(monic=True)  # random

Use degree to obtain polynomials of higher degree:

Sage

sage: # needs sage.rings.finite_rings
sage: p = S.random_element(degree=5)   # random
(t^2 + 3*t)*x^5 + (4*t + 4)*x^3 + (4*t^2 + 4*t)*x^2 + (2*t^2 + 1)*x + 3
sage: p.degree() == 5
True

Python

>>> from sage.all import *
>>> # needs sage.rings.finite_rings
>>> p = S.random_element(degree=Integer(5))   # random
(t^2 + 3*t)*x^5 + (4*t + 4)*x^3 + (4*t^2 + 4*t)*x^2 + (2*t^2 + 1)*x + 3
>>> p.degree() == Integer(5)
True

Sage Live

# needs sage.rings.finite_rings
p = S.random_element(degree=5)   # random
p.degree() == 5

If a tuple of two integers is given for the degree argument, a random integer will be chosen between the first and second element of the tuple as the degree, both inclusive:

Sage

sage: S.random_element(degree=(2,7))  # random                              # needs sage.rings.finite_rings
(3*t^2 + 1)*x^4 + (4*t + 2)*x^3 + (4*t + 1)*x^2
 + (t^2 + 3*t + 3)*x + 3*t^2 + 2*t + 2

Python

>>> from sage.all import *
>>> S.random_element(degree=(Integer(2),Integer(7)))  # random                              # needs sage.rings.finite_rings
(3*t^2 + 1)*x^4 + (4*t + 2)*x^3 + (4*t + 1)*x^2
 + (t^2 + 3*t + 3)*x + 3*t^2 + 2*t + 2

Sage Live

S.random_element(degree=(2,7))  # random                              # needs sage.rings.finite_rings

random_irreducible(degree=2, monic=True, *args, **kwds)[source]¶

Return a random irreducible Ore polynomial.

Warning

Elements of this Ore polynomial ring need to have a method is_irreducible(). Currently, this method is implemented only when the base ring is a finite field.

INPUT:

degree – integer with degree (default: 2) or a tuple of integers with minimum and maximum degrees
monic – if True, returns a monic Ore polynomial (default: True)
*args, **kwds – passed in to the random_element method for the base ring

EXAMPLES:

Sage

sage: # needs sage.rings.finite_rings
sage: k.<t> = GF(5^3)
sage: Frob = k.frobenius_endomorphism()
sage: S.<x> = k['x',Frob]
sage: A = S.random_irreducible()
sage: A.is_irreducible()
True
sage: B = S.random_irreducible(degree=3, monic=False)
sage: B.is_irreducible()
True

Python

>>> from sage.all import *
>>> # needs sage.rings.finite_rings
>>> 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)
>>> A = S.random_irreducible()
>>> A.is_irreducible()
True
>>> B = S.random_irreducible(degree=Integer(3), monic=False)
>>> B.is_irreducible()
True

Sage Live

# needs sage.rings.finite_rings
k.<t> = GF(5^3)
Frob = k.frobenius_endomorphism()
S.<x> = k['x',Frob]
A = S.random_irreducible()
A.is_irreducible()
B = S.random_irreducible(degree=3, monic=False)
B.is_irreducible()

twisting_derivation()[source]¶

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

EXAMPLES:

Sage

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

Python

>>> 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)
>>> A.twisting_derivation()
d/dt

>>> # needs sage.rings.finite_rings
>>> 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)
>>> S.twisting_derivation()

Sage Live

R.<t> = QQ[]
der = R.derivation(); der
A.<d> = R['d', der]
A.twisting_derivation()
# needs sage.rings.finite_rings
k.<a> = GF(5^3)
Frob = k.frobenius_endomorphism()
S.<x> = k['x', Frob]
S.twisting_derivation()