Deformation of roots of polynomials via fractional derivatives

We 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 Pf(x,q). For 0⩽q⩽1, Pf(x,q) defines a homotopy between the polynomials f(x) and xf′(x). Iterating this construction, we associate to f(x) a plane spline curve, called the stem of f. Stems of classic random polynomials exhibits intriguing patterns; moreover in the complex plane Pf(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.