def center(self, name=None, names=None, default=False):
r"""
Return the center of this skew polynomial ring.
.. NOTE::
If `F` denotes the subring of `R` fixed by `\sigma` and `\sigma`
has order `r`, the center of `K[x,\sigma]` is `F[x^r]`, that
is a univariate polynomial ring over `F`.
INPUT:
- ``name`` -- a string or ``None`` (default: ``None``);
the name for the central variable (namely `x^r`)
- ``default`` -- a boolean (default: ``False``); if ``True``,
set the default variable name for the center to ``name``
EXAMPLES::
sage: k.<t> = GF(5^3)
sage: Frob = k.frobenius_endomorphism()
sage: S.<x> = k['x',Frob]; S
Ore Polynomial Ring in x over Finite Field in t of size 5^3 twisted by t |--> t^5
sage: Z = S.center(); Z
Univariate Polynomial Ring in z over Finite Field of size 5
sage: Z.gen()
z
We can pass in another variable name::
sage: S.center(name='y')
Univariate Polynomial Ring in y over Finite Field of size 5
or use the bracket notation::
sage: Zy.<y> = S.center(); Zy
Univariate Polynomial Ring in y over Finite Field of size 5
sage: y.parent() is Zy
True
A coercion map from the center to the skew polynomial ring is set::
sage: S.has_coerce_map_from(Zy)
True
sage: P = y + x; P
x^3 + x
sage: P.parent()
Ore Polynomial Ring in x over Finite Field in t of size 5^3 twisted by t |--> t^5
sage: P.parent() is S
True
together with a conversion map in the reverse direction::
sage: Zy(x^6 + 2*x^3 + 3)
y^2 + 2*y + 3
sage: Zy(x^2)
Traceback (most recent call last):
...
ValueError: x^2 is not in the center
Two different skew polynomial rings can share the same center::
sage: S1.<x1> = k['x1', Frob]
sage: S2.<x2> = k['x2', Frob]
sage: S1.center() is S2.center()
True
.. RUBRIC:: About the default name of the central variable
A priori, the default is ``z``.
However, a variable name is given the first time this method is
called, the given name become the default for the next calls::
sage: K.<t> = GF(11^3)
sage: phi = K.frobenius_endomorphism()
sage: A.<X> = K['X', phi]
sage: C.<u> = A.center() # first call
sage: C
Univariate Polynomial Ring in u over Finite Field of size 11
sage: A.center() # second call: the variable name is still u
Univariate Polynomial Ring in u over Finite Field of size 11
sage: A.center() is C
True
We can update the default variable name by passing in the argument
``default=True``::
sage: D.<v> = A.center(default=True)
sage: D
Univariate Polynomial Ring in v over Finite Field of size 11
sage: A.center()
Univariate Polynomial Ring in v over Finite Field of size 11
sage: A.center() is D
True
TESTS::
sage: C.<a,b> = S.center()
Traceback (most recent call last):
...
IndexError: the number of names must equal the number of generators
"""
if name is not None and names is not None:
raise ValueError("you must specify the name of the variable")
if names is None:
if name is None:
name = self._center_variable_name
if name is None:
name = 'z'
names = (name, )
names = normalize_names(1, names)
name = names[0]
if name in self._center:
center = self._center[name]
else:
center = PolynomialRing(self._constants, names)
embed = SkewPolynomialCenterInjection(center, self,
self._embed_constants,
self._order)
try:
assert not self.has_coerce_map_from(center)
self.register_coercion(embed)
center.register_conversion(embed.section())
except AssertionError:
raise ValueError(
"creation of coercion map fails; consider using another variable name"
)
self._center[name] = center
if default or (self._center_variable_name is None):
self._center_variable_name = name
return center
