Virtual Roots of a Real Polynomial and Fractional Derivatives

After the works of Gonzales-Vega, Lombardi, Mahé,\cite{Lomb1} and Coste, Lajous, Lombardi, Roy \cite{Lomb2}, we consider the virtual roots of a univariate polynomial $f$ with real coefficients. Using fractional derivatives, we associate to $f$ a bivariate polynomial $P_f(x,t)$ depending on the choice of an origin $a$, then two type of plan curves we call the FDcurve and stem of $f$. We show, in the generic case, how to locate the virtual roots of $f$ on the Budan table and on each of these curves. The paper is illustrated with examples and pictures computed with the computer algebra system Maple.