Virtual Roots of Real Polynomials

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.