  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.
