New interface

# Deformation of Roots of Polynomials via Fractional Derivatives

2 GALAAD - Geometry, algebra, algorithms
CRISAM - Inria Sophia Antipolis - Méditerranée , UNS - Université Nice Sophia Antipolis (1965 - 2019), 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.
Keywords :
Document type :
Journal articles

Cited literature [16 references]

https://hal.inria.fr/hal-00653770
Contributor : André Galligo Connect in order to contact the contributor
Submitted on : Tuesday, December 20, 2011 - 11:17:54 AM
Last modification on : Thursday, August 4, 2022 - 5:05:33 PM
Long-term archiving on: : Wednesday, March 21, 2012 - 2:25:25 AM

### File

RootsDeformationGalligo.pdf.pd...
Files produced by the author(s)

### Citation

André Galligo. Deformation of Roots of Polynomials via Fractional Derivatives. Journal of Symbolic Computation, 2013, 52, pp.35-50. ⟨10.1016/j.jsc.2012.05.011⟩. ⟨hal-00653770⟩

Record views