Deformation of Roots of Polynomials via Fractional Derivatives

International audienceWe first recall the main features of Fractional calculus. In the expression of fractional derivatives of a real polynomial $f(x)$, we view the order of differentiation $q$ as a new indeterminate; then we define a new bivariate polynomial $P_f(x,q)$. For $0 \leq q \leq 1$, $P_f(x,q)$ defines an homotopy between the polynomials $f(x)$ and $xf'(x)$. Iterating this construction, we associate to $f(x)$ a plane spline curve, we called the stem of $f$. Stems of classic random polynomials exhibits intriguing patterns; moreover in the complex plane $P_f(x,q)$ creates an unexpected correspondence between the complex roots and the critical points of $f(x)$. We propose 3 conjectures to describe and explain these phenomena. Illustrations are provided relying on the computer algebra system Maple.On utilise un facteur polynomial de la derivee fractionnaire d'un polynome f pour definir une homotopie entre ce polynome et son polynome derive. Ceci permet d'associer a f deux type de courbes planes. On etudie le cas ou les coefficients de f suivent des distributions aleatoires classiques