def conj(P):
"""
Conjugate of a polynomial P.
"""
Pc = P.coefficients()
x = P.parent().gen()
return sum([Pc[p] * generic_power(x,p) for p in range(0,len(Pc))\
if Pc[p].imag() == 0]) - \
sum([Pc[p] * generic_power(x,p) for p in range(0,len(Pc)) \
if Pc[p].imag() != 0])
def _pow_int(self, n):
"""
Return ``self`` to the `n^{th}` power.
INPUT:
- ``n`` -- a positive integer
EXAMPLES::
sage: S = Semigroups().example("leftzero")
sage: x = S("x")
sage: x^1, x^2, x^3, x^4, x^5
('x', 'x', 'x', 'x', 'x')
sage: x^0
Traceback (most recent call last):
...
ArithmeticError: only positive powers are supported in a semigroup
TESTS::
sage: x._pow_int(17)
'x'
"""
if n <= 0:
raise ArithmeticError("only positive powers are supported in a semigroup")
return generic_power(self, n)
def realpart(P):
"""
Real part of a polynomial P.
(ie: compute sum P_i x^i only for P_i real).
"""
x = P.parent().gen()
Pc = P.coefficients()
return sum([Pc[p] * generic_power(x,p) for p in range(0,len(Pc)) \
if Pc[p].imag() == 0])
def impart(P):
"""
Imaginary part of a polynomial P.
(ie: compute sum Im P_i x^i only for P_i imaginary).
"""
Pc = P.coefficients()
x = P.parent().gen()
Im = QQbar(I)
return sum([-Im * Pc[p]*generic_power(x,p) \
for p in range(0,len(Pc)) if Pc[p].imag() != 0])
def _pow_int(self, n):
r"""
Return ``self`` to the `n^{th}` power.
INPUT:
- ``n`` -- a nonnegative integer
EXAMPLES::
sage: S = Monoids().example()
sage: S("a") ^ 5
'aaaaa'
"""
return generic_power(self, n)
def _pow_int(self, n):
r"""
Return ``self`` to the `n^{th}` power.
INPUT:
- ``n`` -- a nonnegative integer
EXAMPLES::
sage: S = Monoids().example()
sage: S("a") ^ 5
'aaaaa'
"""
return generic_power(self, n)
def __pow__(self, n):
"""
Return the `n^{th}` power of a factorization, which is got by
combining together like factors.
EXAMPLES::
sage: f = factor(-100); f
-1 * 2^2 * 5^2
sage: f^3
-1 * 2^6 * 5^6
sage: f^4
2^8 * 5^8
sage: K.<a> = NumberField(x^3 - 39*x - 91)
sage: F = K.factor(7); F
(Fractional ideal (7, a)) * (Fractional ideal (7, a + 2)) * (Fractional ideal (7, a - 2))
sage: F^9
(Fractional ideal (7, a))^9 * (Fractional ideal (7, a + 2))^9 * (Fractional ideal (7, a - 2))^9
sage: R.<x,y> = FreeAlgebra(ZZ, 2)
sage: F = Factorization([(x,3), (y, 2), (x,1)]); F
x^3 * y^2 * x
sage: F**2
x^3 * y^2 * x^4 * y^2 * x
"""
from sage.rings.integer import Integer
if not isinstance(n, Integer):
try:
n = Integer(n)
except TypeError:
raise TypeError("Exponent n (= %s) must be an integer." % n)
if n == 1:
return self
if n == 0:
return Factorization([])
if self.is_commutative():
return Factorization([(p, n * e) for p, e in self],
unit=self.unit()**n,
cr=self.__cr,
sort=False,
simplify=False)
if n < 0:
self = ~self
n = -n
from sage.arith.power import generic_power
return generic_power(self, n)
def __pow__(self, n):
"""
Return the `n^{th}` power of a factorization, which is got by
combining together like factors.
EXAMPLES::
sage: f = factor(-100); f
-1 * 2^2 * 5^2
sage: f^3
-1 * 2^6 * 5^6
sage: f^4
2^8 * 5^8
sage: K.<a> = NumberField(x^3 - 39*x - 91)
sage: F = K.factor(7); F
(Fractional ideal (7, a)) * (Fractional ideal (7, a + 2)) * (Fractional ideal (7, a - 2))
sage: F^9
(Fractional ideal (7, a))^9 * (Fractional ideal (7, a + 2))^9 * (Fractional ideal (7, a - 2))^9
sage: R.<x,y> = FreeAlgebra(ZZ, 2)
sage: F = Factorization([(x,3), (y, 2), (x,1)]); F
x^3 * y^2 * x
sage: F**2
x^3 * y^2 * x^4 * y^2 * x
"""
if not isinstance(n, Integer):
try:
n = Integer(n)
except TypeError:
raise TypeError("Exponent n (= %s) must be an integer." % n)
if n == 1:
return self
if n == 0:
return Factorization([])
if self.is_commutative():
return Factorization([(p, n*e) for p, e in self], unit=self.unit()**n, cr=self.__cr, sort=False, simplify=False)
if n < 0:
self = ~self
n = -n
from sage.arith.power import generic_power
return generic_power(self, n)
def __pow__(self, n):
"""
Return the `n`-th power of the series.
EXAMPLES::
sage: L.<z> = LazyLaurentSeriesRing(ZZ)
sage: (1 - z)^-1
1 + z + z^2 + z^3 + z^4 + z^5 + z^6 + ...
sage: (1 - z)^0
1
sage: (1 - z)^3
1 - 3*z + 3*z^2 - z^3
sage: (1 - z)^-3
1 + 3*z + 6*z^2 + 10*z^3 + 15*z^4 + 21*z^5 + 28*z^6 + ...
"""
if n == 0:
return self.parent().one()
return generic_power(self, n)
