Mathematical background

Let $P$ the polynomial:

$$P=a_0+a_1x+\ldots+a_nx^n$$

If all the roots of $P$ are real then [13]:

$$a_i ^2 \ge a_{i-1}a_{i+1}~~~~{\rm for }~ i \in [1,n]$$

Conversely if this relation does not hold for some $i$ (for example if a coefficient is 0 while its neighbor have same sign), then there are complex roots.