Chebyshev's method applied to polynomials with rotational symmetry

Tarakanta Nayak; Pooja Phogat

paper

Chebyshev's method applied to polynomials with rotational symmetry

arXiv:2609.02884

Abstract

We investigate the dynamics of Chebyshev's method applied to the polynomial family $p_{n} (z) = z (z^{n} - 1)$ for $n > 1$ . The resulting map is denoted by $C_{n}$ . It is proved that the immediate basins corresponding to the non-zero roots are unbounded and simply connected. We also show that the Julia set of $C_{n}$ is connected. It is proved that the immediate basin of the root at the origin exhibits a different behavior: it is unbounded for $n \leq 16$ and bounded for $n \geq 17$ . We establish that $C_{n}$ is convergent whenever $n \leq 16$ or $n$ is odd. Finally, we determine the symmetry group of $C_{n}$ and prove that it coincides with the symmetry group of the polynomial $p_{n}$ , thereby confirming, for this family, a conjecture proposed by Nayak and Pal.

30 Pages