Deformation of roots of polynomials via fractional derivatives. (English) Zbl 1261.65046
Summary: 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 a homotopy between the polynomials \(f(x)\) and \(xf^{\prime}(x)\). 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.
MSC:
65H04 | Numerical computation of roots of polynomial equations |
65D07 | Numerical computation using splines |
26A33 | Fractional derivatives and integrals |
12D10 | Polynomials in real and complex fields: location of zeros (algebraic theorems) |
65E05 | General theory of numerical methods in complex analysis (potential theory, etc.) |
30C15 | Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral) |
65H20 | Global methods, including homotopy approaches to the numerical solution of nonlinear equations |
68W30 | Symbolic computation and algebraic computation |