Generalized Budan-Fourier theorem and virtual roots

The notion of virtual root was introduced in [5] in the case of polynomials. The virtual roots provide d continuous “root fonctions” on the space all real polynomials of a given degree d, with an interlacing property linking virtual roots of P and virtual roots of P ′. From a computational point of view, there is no need to know the coefficients with infinite precision in order to compute the virtual roots with finite precision: the discontinuity phenomenon of real roots vanishing in the complex plane “disappears”. Another nice fact is that all real roots of P are virtual roots and all virtual roots are real roots of P or of one of its derivatives. Theorem 2 of this note shows that the Budan-Fourier count always gives the number of virtual roots (with multiplicities) on an interval (a, b]. So the quantity by which the Budan-Fourier count exceeds the number of actual roots is explained by the presence of extra virtual roots.