Hello :) I'm working in a combinatorics problem where I need to describe the roots of the denominator of a generating function, specifically, $p_k(z)=z^{k+2}-z^3+z^2+z-1$, with $(k\geq1)$. The cases $k=1,2$ are solvable by radicals, and I would like to be able to express the roots of $p_k(z)$ in therms of $k$. I don't know too much about Galois Theory, but I found some results:
- By Descartes' rule of signs, $p_k(z)$ has at least one positive real root.
- By Bolzano's Theorem, $p_k(z)$ has at least one positive root on the interval $(0,1)$. Numerically, it seems to be the only positive root, and if $\alpha_k$ is this root then $\alpha_k\to1$ as $k\to\infty$.
I'm interested on the positive root. I'd like to know if there is any way to prove the unicity of $\alpha_k$ and giving a closed expression. Of course, this formula may be not expressable by radicals, but I'm open to use another trascendental functions like we do in the Viète's formula for the depressed cubic.
Let $f_k(x)=x^{k+2}$. Let $g(x)=x^3-x^2-x+1=(x-1)^2(x+1)$. Then $p_k=f_k-g$.
We first show that there is a unique positive root $\alpha_k$ of $p_k$ and $\alpha_k\in (0,1)$ for all $k\ge 1$. The existence can be proved by intermediate value theorem, as noted by OP. For uniqueness, we have $f'_k(x)=(k+2)x^{k+1}$ and $g'(x)=3x^2-2x-1<3x^2$ for all $x>0$. Note that $f_k$ and $-g$ are strictly increasing on $(0,1)$, and so is $p_k$. Then there is only one root of $p_k$ in $(0,1)$. Also, $p_k(1)=1$, so $1$ is not a root. For $x>1$, we have $f_k(1)>g_k(1)$ and $f'_k\ge g'$ on $(1,\infty)$. By racetrack principle, we have $p_k>0$ on $(1,\infty)$. This shows that $\alpha_k$ is the unique root of $p_k$ on $(0,\infty)$.
Now we prove that $\displaystyle\lim_{k\to\infty} \alpha_k=1$. Let $\varepsilon>0$. Note that $g(x)>0$ for all $x\in (0,1)$. Then there exists $N>0$ such that $f_N(1-\varepsilon)<g(1-\varepsilon)$. Let $k>N$. We have $f_k(x)<f_N(x)$ for all $x\in (0,1)$, so $f_k(1-\varepsilon)<g(1-\varepsilon)$. Then $p_k(1-\varepsilon)<0$ and $p_k(1)=1$. By intermediate value theorem and uniqueness of $\alpha_k$, we have $\alpha_k\in (1-\varepsilon, 1)$. This shows that $|1-\alpha_k|<\varepsilon$ for all $k>N$. Thus, $\displaystyle\lim_{k\to \infty} \alpha_k=1$.
As for closed form of $\alpha_k$, it depends on what kind of expressions you are looking for. It most probably does not have a finite radical form. For $1\le k\le 30$, by a Magma computation, the Galois group of $p_k$ is solvable only if $k=1,2,4$. That means $\alpha_1,\alpha_2,\alpha_4$ are the only roots of $p_k$ for $k\le 30$ that can be expressed in finite radical form.