Deformation of Roots of Polynomials via Fractional Derivatives

André Galligo 1, 2
2 GALAAD - Geometry, algebra, algorithms
CRISAM - Inria Sophia Antipolis - Méditerranée , UNS - Université Nice Sophia Antipolis, CNRS - Centre National de la Recherche Scientifique : UMR6621
Abstract : 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 $P_f(x,q)$. 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.
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André Galligo. Deformation of Roots of Polynomials via Fractional Derivatives. Journal of Symbolic Computation, Elsevier, 2013, 52, pp.35-50. ⟨10.1016/j.jsc.2012.05.011⟩. ⟨hal-00653770⟩



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