For a study I'm making about the minimum and maximum radial values of bounded orbits in a central force system with general power law forces, I came across this special polynomial equation:
$${x^\alpha } + {x^{\alpha - 1}} + \ldots + {x^3} + {x^2} -p= 0, \qquad p>0 ,$$
that is
$\sum\limits_{k = 0}^{\alpha - 2} {{x^{\alpha - k}}} -p=0 .$
The polynomial equation is special since:
- all the coefficient of the terms with degree $ \ge 2$ are equal
- the coefficient of the term with degree $1$ is $0$
- the coefficient of the term with degree $0$ is $-p$ with $p>0$
Using Descartes’ rules of sign to count the number of real positive zeros of above equation (and seeing the numerical solutions given by Mathematica for positive values of $p$) I know that above equation has a single real positive solution.
I'm also aware that there aren't, in general, closed algebraic forms for the solutions of polynomial equations with degree $ \ge 5$ but I wonder if, given the special form of the equation, it is possible to express the single real positive root in closed form.
Clearly the algebraic expression for the real positive root would depend on $p$ and $\alpha$.
Even if there's no closed form for the real positive root maybe there are tools for exploring above special equation (I'm no expert in this field) that I don't know of.
Any link/suggestion is much appreciated.
In general, there is no algebraic formula for such a root: In the case $\alpha = 5$, $p = 1$, for example, the Galois group of the resulting polynomial, $$x^5 + x^4 + x^3 + x^2 - 1 ,$$ is the full symmetric group $S_5$. Since this group is not solvable, there is no expression for any of its roots in terms of basic operations and root extraction.
Of course, in some special cases one may still be able to find a formula; for example, when $p = \alpha - 1$, the unique positive solution is $x = 1$.