I am working with the polynomial $x^4+ax^3+bx^2+cx+d$, with $a>0, b>0, c>0, d>0$.
I would like to understand under which conditions, on the coefficients, this polynomial has only complex roots with negative real part or real negative roots.
Does someone knows this conditions or someone has some literature to suggest?
Answer
Given a polynomial $P=a_0 x^n + a_1 x^{n-1} + \cdots + a_n$, set up the $(n+1)\times (n+1)$ Hurwitz matrix
$$H(P)= \begin{matrix} a_0 & a_2 & \ldots &\\
0& a_1 & a_3 & \ldots\\
0& a_0 & a_2 & \ldots\\
0 & 0 & a_1 & \ldots \\
0& 0 & \ldots & a_{n+1} \end{matrix} $$
$H(P)$ is constructed as follows: on the first row we place $a$'s with even indexes. Then, in each column, the indexes decrease by $1$. On the diagonal we have the coefficients $a_0$, $\ldots$, $a_{n+1}$. Note that $a_{\mu}$ is $0$ for $\mu>n$, or for $\mu<0$.
Assume that $a_0>0$. Then $P$ has all roots with real part $<0$ if and only if all the leading minors of the matrix $H(P)$ are $>0$. Note that this is equivalent to $a_0>0$, $a_n>0$ and the principal minors $(2,\ldots,k)$ positive for all $2\le k \le n$, that is, $a_0$, $a_1$, $a_n>0$ and $n-2$ other algebraic conditions.
In the case $n=4$, appearing also in Hurwitz original paper, we have
$$P= a_0 x^4 + a_1 x^3 + a_2 x^2 + a_3 x + a_4$$
the Hurwitz matrix is
$$H(P) = \left(\begin{matrix} a_0 & a_2 & a_4 & 0 & 0\\
0& a_1 & a_3 & 0 & 0\\
0& a_0& a_2 & a_4 & 0\\
0& 0& a_1 & a_3 & 0\\
0& 0& a_0& a_2 & a_4\end{matrix} \right)$$
and the conditions, on top of $a_i>0$, are
$$\left| \begin{matrix} a_1 & a_3 \\
a_0 & a_2 \end{matrix} \right |>0 \\
\left | \begin{matrix} a_1 & a_3 & 0\\
a_0 & a_2 & a_4\\
0& a_1 & a_3 \end{matrix} \right |>0$$
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