Deformation of roots of polynomials via fractional derivatives

作者：Galligo Andre^*

来源：Journal of Symbolic Computation, 2013, 52: 35-50.

DOI：10.1016/j.jsc.2012.05.011

• Fractional derivatives
• Deformation
• Roots of real univariate polynomials
• Complex critical points
• Random polynomials
• Patterns
• FD-curve
• Stems

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 %26lt;= q %26lt;= 1, P-f(x, q) defines a homotopy between the polynomials f(x) and xf%26apos;(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 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.

• 出版日期2013-5