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Merge branch 'skew_polynomial_finite_order' into skew_polynomial_fini…
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xcaruso committed Apr 8, 2020
2 parents 86b0bdc + 49c64bb commit 1b0092f
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5 changes: 0 additions & 5 deletions src/sage/rings/polynomial/skew_polynomial_element.pyx
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Original file line number Diff line number Diff line change
Expand Up @@ -77,11 +77,6 @@ from sage.rings.polynomial.polynomial_element cimport _dict_to_list
from sage.structure.element import coerce_binop
from sage.misc.superseded import experimental

cdef class CenterSkewPolynomial_generic_dense(Polynomial_generic_dense):
"""
A class for elements in the center of a skew polynomial ring.
"""
pass

cdef class SkewPolynomial(AlgebraElement):
r"""
Expand Down
149 changes: 71 additions & 78 deletions src/sage/rings/polynomial/skew_polynomial_finite_order.pyx
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Original file line number Diff line number Diff line change
Expand Up @@ -169,21 +169,26 @@ cdef class SkewPolynomial_finite_order_dense(SkewPolynomial_generic_dense):
sage: k.<t> = GF(5^3)
sage: Frob = k.frobenius_endomorphism()
sage: S.<x> = k['x',Frob]
sage: a = S.random_element(degree=3,monic=True); a #random
x^3 + (2*t^2 + 3)*x^2 + (4*t^2 + t + 4)*x + 2*t^2 + 2
sage: T = a.reduced_trace(); T #random
3*(x^3) + 4
Note that the parent of `T` is the center of the `S`
(and not `S` itself)::
sage: T.parent()
Center of Skew Polynomial Ring in x over Finite Field in t of size 5^3 twisted by t |--> t^5:
Univariate Polynomial Ring in (x^3) over Finite Field of size 5
sage: T.parent() == S.center()
sage: a = x^3 + (2*t^2 + 3)*x^2 + (4*t^2 + t + 4)*x + 2*t^2 + 2
sage: tr = a.reduced_trace(); tr #random
3*z + 4
The reduced trace lies in the center of `S`, which is a univariate
polynomial ring in the variable `z = x^3` over `GF(5)`.
sage: tr.parent()
Univariate Polynomial Ring in z over Finite Field of size 5
sage: tr.parent() is S.center()
True
We check that the reduced trace is additive:
We can use explicit conversion to view ``tr`` as a skew polynomial::
sage: S(tr)
3*x^3 + 4
TESTS:
We check that the reduced trace is additive::
sage: a = S.random_element(degree=5)
sage: b = S.random_element(degree=7)
Expand All @@ -198,8 +203,8 @@ cdef class SkewPolynomial_finite_order_dense(SkewPolynomial_generic_dense):
for _ in range(order-1):
tr = c + twist_map(tr)
coeffs.append(tr)
return self.parent().center()(coeffs)

Z = self.parent().center()
return Z(coeffs)

def reduced_norm(self):
r"""
Expand All @@ -209,46 +214,27 @@ cdef class SkewPolynomial_finite_order_dense(SkewPolynomial_generic_dense):
The result is cached.
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 (considered as a polynomial of the variable `X^r`)
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`).
EXAMPLES::
sage: k.<t> = GF(5^3)
sage: Frob = k.frobenius_endomorphism()
sage: S.<x> = k['x',Frob]
sage: a = S.random_element(degree=3,monic=True); a
x^3 + (2*t^2 + 3)*x^2 + (4*t^2 + t + 4)*x + 2*t^2 + 2
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
(x^3)^3 + 4*(x^3)^2 + 4
z^3 + 4*z^2 + 4
Note that the parent of `N` is the center of the `S`
(and not `S` itself)::
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: N.parent()
Center of Skew Polynomial Ring in x over Finite Field in t of size 5^3 twisted by t |--> t^5:
Univariate Polynomial Ring in (x^3) over Finite Field of size 5
sage: N.parent() == S.center()
Univariate Polynomial Ring in z over Finite Field of size 5
sage: N.parent() is S.center()
True
In any case, coercion works fine::
We can use explicit conversion to view ``tr`` as a skew polynomial::
sage: S(N)
x^9 + 4*x^6 + 4
sage: N + a
x^9 + 4*x^6 + x^3 + (2*t^2 + 3)*x^2 + (4*t^2 + t + 4)*x + 2*t^2 + 1
We check that `N` is a multiple of `a`::
Expand All @@ -257,29 +243,34 @@ cdef class SkewPolynomial_finite_order_dense(SkewPolynomial_generic_dense):
sage: S(N).is_left_divisible_by(a)
True
.. NOTE::
We really need to coerce first `N` into `S`. Otherwise an
error occurs::
sage: N.is_right_divisible_by(a)
Traceback (most recent call last):
...
AttributeError: 'sage.rings.polynomial.skew_polynomial_element.CenterSkewPolynomial_generic_dense' object has no attribute 'is_right_divisible_by'
We check that the reduced norm is a multiplicative map::
sage: a = S.random_element(degree=5)
sage: b = S.random_element(degree=7)
sage: a.reduced_norm() * b.reduced_norm() == (a*b).reduced_norm()
True
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`).
"""
if self._norm is None:
center = self.parent().center()
if self.is_zero():
self._norm = center(0)
else:
section = center._embed_basering.section()
section = self._parent._embed_constants.section()
exp = (self.parent().base_ring().cardinality() - 1) / (center.base_ring().cardinality() - 1)
order = self.parent()._order
lc = section(self.leading_coefficient()**exp)
Expand Down Expand Up @@ -345,20 +336,20 @@ cdef class SkewPolynomial_finite_order_dense(SkewPolynomial_generic_dense):
sage: k.<t> = GF(5^3)
sage: Frob = k.frobenius_endomorphism()
sage: S.<x> = k['x',Frob]
sage: Z = S.center()
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
(x^3)^2 + (x^3) + 4
z^2 + z + 4
Note that the parent of `b` is the center of the `S`
(and not `S` itself)::
We observe that the bound is explicity 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: b.parent()
Center of Skew Polynomial Ring in x over Finite Field in t of size 5^3 twisted by t |--> t^5:
Univariate Polynomial Ring in (x^3) over Finite Field of size 5
sage: b.parent() == Z
True
sage: S(b)
x^6 + x^3 + 4
We check that `b` is divisible by `a`::
Expand All @@ -376,15 +367,17 @@ cdef class SkewPolynomial_finite_order_dense(SkewPolynomial_generic_dense):
it affects the behaviour of ``bound()``::
sage: a.optimal_bound()
(x^3) + 3
z + 3
sage: a.bound()
(x^3) + 3
z + 3
We finally check that if `a` is a central skew polynomial,
then ``a.bound()`` returns simply `a`::
TESTS:
sage: a = S(Z.random_element(degree=4))
sage: b = a.bound()
We check that when the input skew polynomial lies in
the center, the output is the skew polynomial itself::
sage: a = Z.random_element(degree=4)
sage: b = S(a).bound()
sage: a == b
True
"""
Expand Down Expand Up @@ -413,20 +406,20 @@ cdef class SkewPolynomial_finite_order_dense(SkewPolynomial_generic_dense):
sage: k.<t> = GF(5^3)
sage: Frob = k.frobenius_endomorphism()
sage: S.<x> = k['x',Frob]
sage: Z = S.center()
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
(x^3) + 3
z + 3
Note that the parent of `b` is the center of the `S`
(and not `S` itself)::
We observe that the bound is explicity 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: b.parent()
Center of Skew Polynomial Ring in x over Finite Field in t of size 5^3 twisted by t |--> t^5:
Univariate Polynomial Ring in (x^3) over Finite Field of size 5
sage: b.parent() == Z
True
sage: S(b)
x^3 + 3
We check that `b` is divisible by `a`::
Expand All @@ -441,11 +434,11 @@ cdef class SkewPolynomial_finite_order_dense(SkewPolynomial_generic_dense):
self._optbound = center(self).monic()
except ValueError:
bound = self._matphir_c().minimal_polynomial()
section = center._embed_basering.section()
section = self._parent._embed_constants.section()
self._optbound = [ section(x) for x in bound.list() ]
return center(self._optbound)


# TODO:
# fast multiplication
# reduced trace, reduced characteristic polynomial
# reduced characteristic polynomial
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