VIRTUAL ROOTS OF REAL POLYNOMIALS

The fact that a real univariate polynomial misses some real roots is u sually overcome by considering complex roots, but the price to pay for , is a complete loss 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 roo ts which keep sign properties and which extend of course the existing roots. Moreover these ''virtual roots'' are the values of semialgebrai c continuous - rather uniformly - functions defined on the set of moni c polynomials. We present some applications. (C) 1998 Elsevier Science B.V.