Mathematical background

Let $P(x)$ be an univariate polynomial of degree $n$:

$$P(x)= x^n + a_{1} x^{n-1}+.....a_n=0$$

with $a_n \not=0$. Let $a_{m_1},a_{m_2},\ldots$ with $m_1>m_2>\ldots$ the $k$ strictly negative coefficients of $P$. Then all the positive real roots of $P$ verify [13]:

$$x \le {\rm Max}\{ (k\vert a_{m_1}\vert)^{1/m_1},(k\vert a_{m_2}\vert)^{1/m_2},\ldots\}$$

Note that if $k=0$ all the roots are negative according to Descartes Lemma (see section 5.5.1).