Positive root of a sparse polynomial

Question

Given $a, b, c > 0$, is it possible to compute the real positive root of the following sparse polynomial of degree $n \geq 2$?

$$P(x) = ax^n - bx + c$$

Answer 1

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.