Mathematical background

Let $P$ the polynomial:

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

with $a_0a_n \not=0$. Let the sequence $\{a_0,a_1,\ldots,a_n\}$ and the $n$ the number of change of sign in this sequence. Then the number of positive real roots, counted with their order of multiplicity is equal to $n-2k$ with $k\in [0,n/2]$ [13].