Positive root of a sparse polynomial Given $a, b, c > 0$, is it possible to compute the real positive root of the following sparse polynomial of degree $n \geq 2$?
$$P(x) = ax^n - bx + c$$
 A: Caveat: In the given setup, it is possible for the polynomial to have no real roots (when $c$ is sufficiently large).  Here, I am assuming that this polynomial has a root.  To check this, we can plug in the minimum from the derivative and check the sign of $f$ at that point.
It depends on what you mean by "compute."  Do you want a closed-form form of the roots?  Do you want to approximate the roots?  For sparse systems, you can sometimes get something out of the Newton polytope, but I'll use a more elementary approach.
First, consider the derivative of $f$.
$$
f'(x)=nax^{n-1}-b.
$$
We can easily see that this has a zero at
$$
x=\left(\frac{b}{na}\right)^{\frac{1}{n-1}}.
$$
This is a minimum of the polynomial and the value of the function of this point must be negative (if the polynomial has any roots - note that in your setup, the polynomial might not have any roots if $c$ is too large).
On the other hand, when $ax^n-bx=0$, the function is positive.  In other words, when 
$$
x=\left(\frac{b}{a}\right)^{\frac{1}{n-1}},
$$
the function is equal to $c$.  
Therefore, the root of interest is within the interval
$$
\left[\left(\frac{b}{na}\right)^{\frac{1}{n-1}},\left(\frac{b}{a}\right)^{\frac{1}{n-1}}\right].
$$
For $n$ large with respect to $\frac{b}{a}$, this is a small interval.
