Virtual roots of real polynomials

with González-Vega L., Mahé L. Journal of Pure and Applied Algebra 124, (1998) 147-166.

here is the pdf file.

see also Generalized Budan-Fourier theorem and virtual roots. (pdf file), dvi file.
with Coste M., Roy M.-F., Lajous T. Journal of Complexity 21 (2005), 479-486.
In this Note we give a proof of a generalized version of the classical Budan-Fourier theorem, interpreting sign variations in the derivatives in terms of virtual roots.

Résumé : Nous définissons dans cet article un substitut réel pour les racines d'un polynome réel lorsqu'elles se sont évanouies dans le plan complexe. Ces racines virtuelles sont des fonctions semi-algébriques continues des coefficients du polynome (supposé unitaire). Nous étudions quelques propriétés de base de ces racines virtuelles et nous présentons quelques applications de cette notion.

Abstract : The fact that a real univariate polynomial misses some real roots is usually overcame by considering complex roots, but the price to pay for, is a complete lost of the sign structure that a set of real roots is endowed with (mutual position on the line, signs of the derivatives, etc...). In this paper we present real substitutes for these missing roots which keep sign properties and which extend of course the existing roots. Moreover these "virtual roots" are the values of semialgebraic continuous -- rather uniformly -- functions defined on the set of monic polynomials. We present some applications.