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 polynom...
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Veröffentlicht in: | Journal of symbolic computation 2013-05, Vol.52, p.35-50 |
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Format: | Artikel |
Sprache: | eng |
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Zusammenfassung: | 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. |
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ISSN: | 0747-7171 1095-855X |
DOI: | 10.1016/j.jsc.2012.05.011 |