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Virtual roots of a real polynomial and fractional derivatives

After the works of Gonzales-Vega, Lombardi, Mahe [11], and of Coste, Lajous, Lombardi, Roy [6], we consider the virtual roots of a univariate polynomial f with real coefficients. Using fractional derivatives, we associate to f a bivariate polynomial P(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.