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Consider the following two polynomials \begin{align} p(s)&:=s^n+\alpha_{n-1}s^{n-1}+\cdots+\alpha_1s+\alpha_0,\\ q(s)&:=s^{n-1}+\alpha_{n-1}s^{n-2}+\cdots+\alpha_2 s+\alpha_1, \end{align} where $\alpha_{0},\dots,\alpha_{n-1}$ are positive real numbers.

Assume that $p(s)=sq(s)+\alpha_0$ is Hurwitz, i.e., every root $\lambda_i\in\mathbb{C}$ of $p(s)$ satisfies $\Re\mathrm{e}(\lambda_i)\leq 0$. Then, can we conclude that $q(s)$ is also Hurwitz?

For $n=2$ and $n=3$, the answer is positive. Indeed, the result follows by applying the Descartes' rule of signs. What about the general case $n>3$?

Any help will be appreciated.

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