Univariate dense skew polynomials over a field with a finite order automorphism¶

AUTHOR:

- Xavier Caruso (2012-06-29): initial version

Arpit Merchant (2016-08-04): improved docstrings, fixed doctests and refactored classes and methods

class sage.rings.polynomial.skew_polynomial_finite_order.SkewPolynomial_finite_order_dense[source]¶

Bases: SkewPolynomial_generic_dense

This method constructs a generic dense skew polynomial over a field equipped with an automorphism of finite order.

INPUT:

parent – parent of self
x – list of coefficients from which self can be constructed
check – flag variable to normalize the polynomial
construct – boolean (default: False)

bound()[source]¶

Return a bound of this skew polynomial (i.e. a multiple of this skew polynomial lying in the center).

Note

Since \(b\) is central, it divides a skew polynomial on the left iff it divides it on the right

ALGORITHM:

Sage first checks whether self is itself in the center. It if is, it returns self
If an optimal bound was previously computed and cached, Sage returns it
Otherwise, Sage returns the reduced norm of self

As a consequence, the output of this function may depend on previous computations (an example is given below).

EXAMPLES:

Sage

sage: k.<t> = GF(5^3)
sage: Frob = k.frobenius_endomorphism()
sage: S.<x> = k['x', Frob]
sage: Z = S.center(); Z
Univariate Polynomial Ring in z over Finite Field of size 5

sage: a = x^2 + (4*t + 2)*x + 4*t^2 + 3
sage: b = a.bound(); b
z^2 + z + 4

Python

>>> 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)
>>> Z = S.center(); Z
Univariate Polynomial Ring in z over Finite Field of size 5

>>> a = x**Integer(2) + (Integer(4)*t + Integer(2))*x + Integer(4)*t**Integer(2) + Integer(3)
>>> b = a.bound(); b
z^2 + z + 4

We observe that the bound is explicitly given as an element of the center (which is a univariate polynomial ring in the variable \(z\)). We can use conversion to send it in the skew polynomial ring:

Sage

sage: S(b)
x^6 + x^3 + 4

Python

>>> from sage.all import *
>>> S(b)
x^6 + x^3 + 4

We check that \(b\) is divisible by \(a\):

Sage

sage: S(b).is_right_divisible_by(a)
True
sage: S(b).is_left_divisible_by(a)
True

Python

>>> from sage.all import *
>>> S(b).is_right_divisible_by(a)
True
>>> S(b).is_left_divisible_by(a)
True

Actually, \(b\) is the reduced norm of \(a\):

Sage

sage: b == a.reduced_norm()
True

Python

>>> from sage.all import *
>>> b == a.reduced_norm()
True

Now, we compute the optimal bound of \(a\) and see that it affects the behaviour of bound():

Sage

sage: a.optimal_bound()
z + 3
sage: a.bound()
z + 3

Python

>>> from sage.all import *
>>> a.optimal_bound()
z + 3
>>> a.bound()
z + 3

is_central()[source]¶

Return True if this skew polynomial lies in the center.

EXAMPLES:

Sage

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

sage: x.is_central()
False
sage: (t*x^3).is_central()
False
sage: (x^6 + x^3).is_central()
True

Python

>>> 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)

>>> x.is_central()
False
>>> (t*x**Integer(3)).is_central()
False
>>> (x**Integer(6) + x**Integer(3)).is_central()
True

optimal_bound()[source]¶

Return the optimal bound of this skew polynomial (i.e. the monic multiple of this skew polynomial of minimal degree lying in the center).

Note

The result is cached.

EXAMPLES:

Sage

sage: k.<t> = GF(5^3)
sage: Frob = k.frobenius_endomorphism()
sage: S.<x> = k['x', Frob]
sage: Z = S.center(); Z
Univariate Polynomial Ring in z over Finite Field of size 5

sage: a = x^2 + (4*t + 2)*x + 4*t^2 + 3
sage: b = a.optimal_bound(); b
z + 3

Python

>>> 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)
>>> Z = S.center(); Z
Univariate Polynomial Ring in z over Finite Field of size 5

>>> a = x**Integer(2) + (Integer(4)*t + Integer(2))*x + Integer(4)*t**Integer(2) + Integer(3)
>>> b = a.optimal_bound(); b
z + 3

We observe that the bound is explicitly given as an element of the center (which is a univariate polynomial ring in the variable \(z\)). We can use conversion to send it in the skew polynomial ring:

Sage

sage: S(b)
x^3 + 3

Python

>>> from sage.all import *
>>> S(b)
x^3 + 3

We check that \(b\) is divisible by \(a\):

Sage

sage: S(b).is_right_divisible_by(a)
True
sage: S(b).is_left_divisible_by(a)
True

Python

>>> from sage.all import *
>>> S(b).is_right_divisible_by(a)
True
>>> S(b).is_left_divisible_by(a)
True

reduced_charpoly(var=None)[source]¶

Return the reduced characteristic polynomial of this skew polynomial.

INPUT:

var – string, a pair of strings or None (default: None); the variable names used for the characteristic polynomial and the center

Note

The result is cached.

EXAMPLES:

Sage

sage: k.<t> = GF(5^3)
sage: Frob = k.frobenius_endomorphism()
sage: S.<u> = k['u', Frob]
sage: a = u^3 + (2*t^2 + 3)*u^2 + (4*t^2 + t + 4)*u + 2*t^2 + 2
sage: chi = a.reduced_charpoly()
sage: chi
x^3 + (2*z + 1)*x^2 + (3*z^2 + 4*z)*x + 4*z^3 + z^2 + 1

Python

>>> from sage.all import *
>>> k = GF(Integer(5)**Integer(3), names=('t',)); (t,) = k._first_ngens(1)
>>> Frob = k.frobenius_endomorphism()
>>> S = k['u', Frob]; (u,) = S._first_ngens(1)
>>> a = u**Integer(3) + (Integer(2)*t**Integer(2) + Integer(3))*u**Integer(2) + (Integer(4)*t**Integer(2) + t + Integer(4))*u + Integer(2)*t**Integer(2) + Integer(2)
>>> chi = a.reduced_charpoly()
>>> chi
x^3 + (2*z + 1)*x^2 + (3*z^2 + 4*z)*x + 4*z^3 + z^2 + 1

The reduced characteristic polynomial has coefficients in the center of \(S\), which is itself a univariate polynomial ring in the variable \(z = u^3\) over \(\GF{5}\). Hence it appears as a bivariate polynomial:

Sage

sage: chi.parent()
Univariate Polynomial Ring in x over Univariate Polynomial Ring in z over Finite Field of size 5

Python

>>> from sage.all import *
>>> chi.parent()
Univariate Polynomial Ring in x over Univariate Polynomial Ring in z over Finite Field of size 5

The constant coefficient of the reduced characteristic polynomial is the reduced norm, up to a sign:

Sage

sage: chi[0] == -a.reduced_norm()
True

Python

>>> from sage.all import *
>>> chi[Integer(0)] == -a.reduced_norm()
True

Its coefficient of degree \(\deg(a) - 1\) is the opposite of the reduced trace:

Sage

sage: chi[2] == -a.reduced_trace()
True

Python

>>> from sage.all import *
>>> chi[Integer(2)] == -a.reduced_trace()
True

By default, the name of the variable of the reduced characteristic polynomial is x and the name of central variable is usually z (see center() for more details about this). The user can speciify different names if desired:

Sage

sage: a.reduced_charpoly(var='T')  # variable name for the caracteristic polynomial
T^3 + (2*z + 1)*T^2 + (3*z^2 + 4*z)*T + 4*z^3 + z^2 + 1

sage: a.reduced_charpoly(var=('T', 'c'))
T^3 + (2*c + 1)*T^2 + (3*c^2 + 4*c)*T + 4*c^3 + c^2 + 1

Python

>>> from sage.all import *
>>> a.reduced_charpoly(var='T')  # variable name for the caracteristic polynomial
T^3 + (2*z + 1)*T^2 + (3*z^2 + 4*z)*T + 4*z^3 + z^2 + 1

>>> a.reduced_charpoly(var=('T', 'c'))
T^3 + (2*c + 1)*T^2 + (3*c^2 + 4*c)*T + 4*c^3 + c^2 + 1

See also

reduced_trace(), reduced_norm()

reduced_norm(var=None)[source]¶

Return the reduced norm of this skew polynomial.

INPUT:

var – string or False or None (default: None); the variable name. If False, return the list of coefficients.

Note

The result is cached.

EXAMPLES:

Sage

sage: k.<t> = GF(5^3)
sage: Frob = k.frobenius_endomorphism()
sage: S.<x> = k['x', Frob]
sage: a = x^3 + (2*t^2 + 3)*x^2 + (4*t^2 + t + 4)*x + 2*t^2 + 2
sage: N = a.reduced_norm(); N
z^3 + 4*z^2 + 4

Python

>>> 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)
>>> a = x**Integer(3) + (Integer(2)*t**Integer(2) + Integer(3))*x**Integer(2) + (Integer(4)*t**Integer(2) + t + Integer(4))*x + Integer(2)*t**Integer(2) + Integer(2)
>>> N = a.reduced_norm(); N
z^3 + 4*z^2 + 4

The reduced norm lies in the center of \(S\), which is a univariate polynomial ring in the variable \(z = x^3\) over \(\GF{5}\):

Sage

sage: N.parent()
Univariate Polynomial Ring in z over Finite Field of size 5
sage: N.parent() is S.center()
True

Python

>>> from sage.all import *
>>> N.parent()
Univariate Polynomial Ring in z over Finite Field of size 5
>>> N.parent() is S.center()
True

We can use explicit conversion to view N as a skew polynomial:

Sage

sage: S(N)
x^9 + 4*x^6 + 4

Python

>>> from sage.all import *
>>> S(N)
x^9 + 4*x^6 + 4

By default, the name of the central variable is usually z (see center() for more details about this). However, the user can specify a different variable name if desired:

Sage

sage: a.reduced_norm(var='u')
u^3 + 4*u^2 + 4

Python

>>> from sage.all import *
>>> a.reduced_norm(var='u')
u^3 + 4*u^2 + 4

When passing in var=False, a tuple of coefficients (instead of an actual polynomial) is returned:

Sage

sage: a.reduced_norm(var=False)
(4, 0, 4, 1)

Python

>>> from sage.all import *
>>> a.reduced_norm(var=False)
(4, 0, 4, 1)

ALGORITHM:

If \(r\) (= the order of the twist map) is small compared to \(d\) (= the degree of this skew polynomial), the reduced norm is computed as the determinant of the multiplication by \(P\) (= this skew polynomial) acting on \(K[X,\sigma]\) (= the underlying skew ring) viewed as a free module of rank \(r\) over \(K[X^r]\).

Otherwise, the reduced norm is computed as the characteristic polynomial of the left multiplication by \(X\) on the quotient \(K[X,\sigma] / K[X,\sigma] P\) (which is a \(K\)-vector space of dimension \(d\)).