I am attempting to take the power series inverse of the Taylor series expansion of $(x^4-x)^{1/3}$ around the point 0.
When I attempt to define g=(x^4-x)^{1/3} :
sage: R.<x> = LaurentPolynomialRing(QQ);
sage: g = (x^4-x)^{1/3}
---------------------------------------------------------------------------
TypeError Traceback (most recent call last)
<ipython-input-21-675538f6f472> in <module>()
----> 1 g = (x**Integer(4)-x)**{Integer(1)/Integer(3)}
sage/rings/polynomial/laurent_polynomial.pyx in sage.rings.polynomial.laurent_polynomial.LaurentPolynomial_univariate.__pow__ (/usr/lib/sagemath//src/build/cythonized/sage/rings/polynomial/laurent_polynomial.c:10802)()
TypeError: int() argument must be a string or a number, not 'set'
Now, when I define $g$ as follows, I am unable to take the power series inverse:
sage: var('x')
x
sage: g = (1-x)**(1/3)
sage: G = g.taylor(x, 0, 10)
sage: 1/G
-4782969/(55913*x^10 + 64515*x^9 + 75735*x^8 + 90882*x^7 + 112266*x^6 + 144342*x^5 + 196830*x^4 + 295245*x^3 + 531441*x^2 + 1594323*x - 4782969)
sage: 1/G + O(x^10)
---------------------------------------------------------------------------
ArithmeticError Traceback (most recent call last)
<ipython-input-12-9fe94787aacb> in <module>()
----> 1 Integer(1)/G + O(x**Integer(10))
/usr/lib/sagemath/local/lib/python2.7/site-packages/sage/rings/big_oh.py in O(*x, **kwds)
154 elif hasattr(x, 'O'):
155 return x.O(**kwds)
--> 156 raise ArithmeticError("O(%s) not defined" % (x,))
ArithmeticError: O(x^10) not defined
sage: G.inverse()
---------------------------------------------------------------------------
AttributeError Traceback (most recent call last)
<ipython-input-14-34018a4209b6> in <module>()
How can I have the best of both worlds? In the end, I would like to have an element in the Laurent Series Ring, for the next stage is to take the formal integral of the result, and the inverse of this formal integral.
