#10720: nth_root in power series
-----------------------------------+----------------------------------------
Reporter: pernici | Owner: pernici
Type: PLEASE CHANGE | Status: new
Priority: minor | Milestone: sage-4.6.2
Component: commutative algebra | Keywords: power series
Author: mario pernici | Upstream: N/A
Reviewer: | Merged:
Work_issues: |
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Comment(by pernici):
The nth-root of power series `y = x^n` is computed using
the Newton method for `y = x^-n`
{{{
x' = (1+1/n)*x - y*x^(n+1)/n
}}}
{{{
sage: R.<t> = QQ[]
sage: p = (1 + 2*t + 5*t^2 + 7*t^3 + O(t^4))^3
sage: p.nth_root(3)
1 + 2*t + 5*t^2 + 7*t^3 + O(t^4)
sage: p = (1 + 2*t + 5*t^2 + 7*t^3 + O(t^4))^-3
sage: p.nth_root(-3)
1 + 2*t + 5*t^2 + 7*t^3 + O(t^4)
}}}
With this patch `nth_root` works also for series on `ZZ`
{{{
sage: R.<t> = ZZ[[]]
sage: p = 1 + 4*t^2 + 3*t^3 + O(t^4)
sage: p.nth_root(2)
1 + 2*t^2 + 3/2*t^3 + O(t^4)
sage: p.sqrt(2)
1 + 2*t^2 + 3/2*t^3 + O(t^4)
sage: p.nth_root(3)
1 + 4/3*t^2 + t^3 + O(t^4)
sage: p = 4*t^2 + 3*t^3 + O(t^4)
sage: p.nth_root(2)
2*t + 3/4*t^2 + O(t^3)
sage: p.sqrt(2)
Traceback (most recent call last):
...
TypeError: no conversion of this rational to integer
}}}
there is a bug in `sqrt` in this case.
`nth_root` can be used to compute the square root of series which
currently `sqrt` does not support
{{{
sage: R.<x,y> = QQ[]
sage: S.<t> = R[[]]
sage: p = 4 + t*x + t^2*y + O(t^3)
sage: p.nth_root(2)
2 + 1/4*x*t + (-1/64*x^2 + 1/4*y)*t^2 + O(t^3)
sage: p = 32*t^5 + x*t^6 + y*t^7 + O(t^8)
sage: p.nth_root(5)
2*t + 1/80*x*t^2 + (-1/6400*x^2 + 1/80*y)*t^3 + O(t^4)
sage: p.sqrt()
Traceback (most recent call last):
...
AttributeError:
'sage.rings.polynomial.multi_polynomial_libsingular.MPolynomial_libsingular'
object has no attribute 'sqrt'
}}}
`nth-root` iteration formula is division-free, which is convenient in this
case; in this case `sqrt` could call `nth_root`.
`nth_root` can be used in the multivariate series considered
in ticket #1956 .
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/10720#comment:3>
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