Virtual Roots of Real Polynomials

International audienceThe fact that a real univariate polynomial misses some real roots is usually over-came by considering complex roots, but the price to pay for, is a complete lost ofthe sign structure that a set of real roots is endowed with (mutual position on theline, signs of the derivatives, etc...). In this paper we present real substitutes forthese missing roots which keep sign properties and which extend of course the exist-ing roots. Moreover these "virtual roots" are the values of semialgebraic continuous{ rather uniformly { functions dened on the set of monic polynomials. We presentsome applications