**Appendix A**

*Appendix A.1. Real Roots Counting of Polynomials*

Consider a monic polynomial of degree *n*

$$f(\mathbf{x}) = \mathbf{x}^n + a\_{n-1}\mathbf{x}^{n-1} + \dots + a\_1\mathbf{x} + a\_0\dots$$

From the fundamental theorem of algebra, it follows that *f* has *n* real or complex roots, counting multiplicities. If the coefficients *a*0, *a*1, ... , *an*−<sup>1</sup> are all real, then the complex roots occur in conjugate pairs.

Using the following Descartes' rules of sign, we can count the number of real positive zeros of *f* . *Descartes' rules*

Let *<sup>p</sup>* be the number of variations in the sign of the coefficients *an*, *an*−1, ... , *<sup>a</sup>*<sup>0</sup> (where *an* = 1 and the zero coefficients are ignored). Let *m* be the number of real positive zeros of *f* . Then,

$$\begin{array}{c} \bullet \qquad m \leq p; \\\hline \end{array}$$

• *p* − *m* is an even integer.

A negative zero of *f*(*x*), if exists, is a positive zero of *f*(−*x*).
