Virtual Roots of a Real Polynomial and Fractional Derivatives

Daniel Bembe 1 André Galligo 2, 3
3 GALAAD - Geometry, algebra, algorithms
CRISAM - Inria Sophia Antipolis - Méditerranée , UNS - Université Nice Sophia Antipolis, CNRS - Centre National de la Recherche Scientifique : UMR6621
Abstract : 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.
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Daniel Bembe, André Galligo. Virtual Roots of a Real Polynomial and Fractional Derivatives. International Symposium on Symbolic and Algebraic Computation (ISSAC), Jun 2011, San Jose, United States. pp.27-34, ⟨10.1145/1993886.1993897⟩. ⟨hal-00559950⟩

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