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fix doctest
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@xcaruso
xcaruso committed Apr 15, 2020
1 parent 8417a05 commit 440ae61
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72 changes: 32 additions & 40 deletions src/sage/rings/polynomial/skew_polynomial_finite_field.pyx
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Original file line number Diff line number Diff line change
Expand Up @@ -48,20 +48,19 @@ from sage.combinat.q_analogues import q_jordan


cdef class SkewPolynomial_finite_field_dense(SkewPolynomial_finite_order_dense):
def _reduced_norm_factor(self):
def _reduced_norm_factored(self):
"""
Return the reduced norm of this polynomial
factorized in the centre.
Return the reduced norm of this polynomial factorized in the center.
EXAMPLES:
sage: k.<t> = GF(5^3)
sage: Frob = k.frobenius_endomorphism()
sage: S.<x> = k['x',Frob]
sage: S.<x> = k['x', Frob]
sage: a = (x^2 + 1) * (x+3)
sage: a.reduced_norm_factor()
((x^3) + 3) * ((x^3) + 2)^2
sage: a._reduced_norm_factored()
(z + 3) * (z + 2)^2
"""
if self._norm_factor is None:
N = self._parent._working_center(self.reduced_norm(var=False))
Expand Down Expand Up @@ -310,7 +309,7 @@ cdef class SkewPolynomial_finite_field_dense(SkewPolynomial_finite_order_dense):
divisors of ``self`` having this norm
"""
skew_ring = self._parent
F = self._reduced_norm_factor()
F = self._reduced_norm_factored()
center = F[0][0].parent()
cardcenter = center.base_ring().cardinality()
gencenter = center.gen()
Expand Down Expand Up @@ -416,7 +415,7 @@ cdef class SkewPolynomial_finite_field_dense(SkewPolynomial_finite_order_dense):
skew_ring = self._parent
center = skew_ring._working_center
kfixed = center.base_ring()
F = self._reduced_norm_factor()
F = self._reduced_norm_factored()
for N,_ in F:
if N == center.gen():
yield skew_ring.gen()
Expand Down Expand Up @@ -498,17 +497,17 @@ cdef class SkewPolynomial_finite_field_dense(SkewPolynomial_finite_order_dense):
We can also generate uniformly distributed irreducible monic
divisors as follows::
sage: a.right_irreducible_divisor(distribution="uniform") # random
sage: a.right_irreducible_divisor(uniform=True) # random
x^3 + (4*t + 2)*x^2 + (2*t^2 + 2*t + 2)*x + 2*t^2 + 2
sage: a.right_irreducible_divisor(distribution="uniform") # random
sage: a.right_irreducible_divisor(uniform=True) # random
x^3 + (t^2 + 2)*x^2 + (3*t^2 + 1)*x + 4*t^2 + 2*t
sage: a.right_irreducible_divisor(distribution="uniform") # random
sage: a.right_irreducible_divisor(uniform=True) # random
x^3 + x^2 + (4*t^2 + 2*t + 4)*x + t^2 + 3
By convention, the zero skew polynomial has no irreducible
divisor:
sage: S(0).irreducible_divisor()
sage: S(0).right_irreducible_divisor()
Traceback (most recent call last):
...
ValueError: 0 has no irreducible divisor
Expand All @@ -518,7 +517,7 @@ cdef class SkewPolynomial_finite_field_dense(SkewPolynomial_finite_order_dense):
if uniform:
N = self._reduced_norm_factor_uniform()
else:
N = self._reduced_norm_factor()[0][0]
N = self._reduced_norm_factored()[0][0]
return self._irreducible_divisor_with_norm(N, True, uniform)

def left_irreducible_divisor(self, uniform=False):
Expand All @@ -527,7 +526,7 @@ cdef class SkewPolynomial_finite_field_dense(SkewPolynomial_finite_order_dense):
if uniform:
N = self._reduced_norm_factor_uniform()
else:
N = self._reduced_norm_factor()[0][0]
N = self._reduced_norm_factored()[0][0]
return self._irreducible_divisor_with_norm(N, False, uniform)


Expand All @@ -542,7 +541,7 @@ cdef class SkewPolynomial_finite_field_dense(SkewPolynomial_finite_order_dense):
sage: Frob = k.frobenius_endomorphism()
sage: S.<x> = k['x',Frob]
sage: a = x^4 + 2*t*x^3 + 3*t^2*x^2 + (t^2 + t + 1)*x + 4*t + 3
sage: iter = a.irreducible_right_divisors(); iter
sage: iter = a.right_irreducible_divisors(); iter
<generator object at 0x...>
sage: next(iter) # random
x + 2*t^2 + 4*t + 4
Expand All @@ -552,25 +551,23 @@ cdef class SkewPolynomial_finite_field_dense(SkewPolynomial_finite_order_dense):
We can use this function to build the list of all monic
irreducible divisors of `a`::
sage: rightdiv = [ d for d in a.irreducible_divisors() ]
sage: rightdiv = [ d for d in a.right_irreducible_divisors() ]
We do some checks::
sage: len(rightdiv) == a.count_irreducible_divisors()
True
sage: len(rightdiv) == len(Set(rightdiv)) # check no duplicates
sage: len(rightdiv) == Set(rightdiv).cardinality() # check no duplicates
True
sage: for d in rightdiv:
... if not a.is_divisible_by(d):
... print "Found %s which is not a right divisor" % d
... elif not d.is_irreducible():
... print "Found %s which is not irreducible" % d
....: assert a.is_right_divisible_by(d), "not right divisible"
....: assert d.is_irreducible(), "not irreducible"
Note that the algorithm is probabilistic. As a consequence, if we
build again the list of right monic irreducible divisors of `a`, we
may get a different ordering::
sage: rightdiv2 = [ d for d in a.irreducible_divisors() ]
sage: rightdiv2 = [ d for d in a.right_irreducible_divisors() ]
sage: rightdiv == rightdiv2
False
sage: Set(rightdiv) == Set(rightdiv2)
Expand Down Expand Up @@ -611,16 +608,14 @@ cdef class SkewPolynomial_finite_field_dense(SkewPolynomial_finite_order_dense):
sage: a = x^4 + (4*t + 3)*x^3 + t^2*x^2 + (4*t^2 + 3*t)*x + 3*t
sage: a.count_irreducible_divisors()
12
sage: a.count_irreducible_divisors(side=Left)
12
We illustrate that an irreducible polynomial in the center have
in general a lot of irreducible divisors in the skew polynomial
ring::
sage: Z = S.center(); x3 = Z.gen()
sage: Z.<x3> = S.center()
sage: N = x3^5 + 4*x3^4 + 4*x3^2 + 4*x3 + 3; N
(x^3)^5 + 4*(x^3)^4 + 4*(x^3)^2 + 4*(x^3) + 3
x3^5 + 4*x3^4 + 4*x3^2 + 4*x3 + 3
sage: N.is_irreducible()
True
sage: S(N).count_irreducible_divisors()
Expand All @@ -631,7 +626,7 @@ cdef class SkewPolynomial_finite_field_dense(SkewPolynomial_finite_order_dense):
skew_ring = self.parent()
cardcenter = skew_ring._working_center.base_ring().cardinality()
gencenter = skew_ring._working_center.gen()
F = self._reduced_norm_factor()
F = self._reduced_norm_factored()
val = self.valuation()
self >>= val
count = 0
Expand Down Expand Up @@ -672,7 +667,7 @@ cdef class SkewPolynomial_finite_field_dense(SkewPolynomial_finite_order_dense):
cdef RingElement unit = <RingElement>self.leading_coefficient()
cdef Polynomial gencenter = skew_ring._working_center.gen()
cdef Py_ssize_t p = skew_ring.characteristic()
cdef F = self._reduced_norm_factor()
cdef F = self._reduced_norm_factored()

for N, m in F:
if N == gencenter:
Expand Down Expand Up @@ -740,7 +735,7 @@ cdef class SkewPolynomial_finite_field_dense(SkewPolynomial_finite_order_dense):
cdef Py_ssize_t m
cdef list type

for N, m in self._reduced_norm_factor():
for N, m in self._reduced_norm_factored():
factorsN += m * [N]
if N == gencenter: continue
type = list(self.type(N))
Expand Down Expand Up @@ -885,11 +880,11 @@ cdef class SkewPolynomial_finite_field_dense(SkewPolynomial_finite_order_dense):
If we rather want uniform distribution among all factorizations,
we need to specify it as follows::
sage: a.factor(distribution="uniform") # random
sage: a.factor(uniform=True) # random
(x + t^2 + 4) * (x + t) * (x + t + 3)
sage: a.factor(distribution="uniform") # random
sage: a.factor(uniform=True) # random
(x + 2*t^2) * (x + t^2 + t + 1) * (x + t^2 + t + 2)
sage: a.factor(distribution="uniform") # random
sage: a.factor(uniform=True) # random
(x + 2*t^2 + 3*t) * (x + 4*t + 2) * (x + 2*t + 2)
By convention, the zero skew polynomial has no factorization:
Expand Down Expand Up @@ -944,7 +939,7 @@ cdef class SkewPolynomial_finite_field_dense(SkewPolynomial_finite_order_dense):
raise ValueError("factorization of 0 not defined")
cardcenter = self._parent._working_center.base_ring().cardinality()
gencenter = self._parent._working_center.gen()
F = self._reduced_norm_factor()
F = self._reduced_norm_factored()
summ = 0
count = 1
for N, m in F:
Expand Down Expand Up @@ -992,15 +987,12 @@ cdef class SkewPolynomial_finite_field_dense(SkewPolynomial_finite_order_dense):
sage: len(factorizations) == a.count_factorizations()
True
sage: len(factorizations) == len(Set(factorizations)) # check no duplicates
sage: len(factorizations) == Set(factorizations).cardinality() # check no duplicates
True
sage: for F in factorizations:
... if F.value() != a:
... print "Found %s which is not a correct factorization" % d
... continue
... for d,_ in F:
... if not d.is_irreducible():
... print "Found %s which is not a correct factorization" % d
....: assert F.value() == a, "factorization has a different value"
....: for d,_ in F:
....: assert d.is_irreducible(), "a factor is not irreducible"
"""
if self.is_zero():
raise ValueError("factorization of 0 not defined")
Expand Down
4 changes: 2 additions & 2 deletions src/sage/rings/polynomial/skew_polynomial_ring.py
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Original file line number Diff line number Diff line change
Expand Up @@ -409,14 +409,14 @@ def __classcall_private__(cls, base_ring, twist_map=None, names=None, sparse=Fal
sage: S is T
True
When the twisting morphism has finite order, a special class
When the twisting morphism is a Frobenius over a finite field, a special class
is used::
sage: k.<a> = GF(7^5)
sage: Frob = k.frobenius_endomorphism(2)
sage: S.<x> = SkewPolynomialRing(k, Frob)
sage: type(S)
<class 'sage.rings.polynomial.skew_polynomial_ring.SkewPolynomialRing_finite_order_with_category'>
<class 'sage.rings.polynomial.skew_polynomial_ring.SkewPolynomialRing_finite_field_with_category'>
"""
if base_ring not in CommutativeRings():
raise TypeError('base_ring must be a commutative ring')
Expand Down

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