PreprintCC BY 4.0arXiv
Chebyshev's method applied to polynomials with rotational symmetry
Abstract
We investigate the dynamics of Chebyshev's method applied to the polynomial family pn(z)=z(zn−1) for n>1. The resulting map is denoted by Cn. 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 Cn is connected. It is proved that the immediate basin of the root at the origin exhibits a different behavior: it is unbounded for n≤ 16 and bounded for n≥ 17. We establish that Cn is convergent whenever n≤ 16 or n is odd. Finally, we determine the symmetry group of Cn and prove that it coincides with the symmetry group of the polynomial pn, thereby confirming, for this family, a conjecture proposed by Nayak and Pal.
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BibTeX
@misc{nayak2026chebyshevs,
title = {{Chebyshev's method applied to polynomials with rotational symmetry}},
author = {Tarakanta Nayak and Pooja Phogat},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.02884},
eprint = {2609.02884},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.02884},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Chebyshev's method applied to polynomials with rotational symmetry
AU - Tarakanta Nayak
AU - Pooja Phogat
PY - 2026
DA - 2026/09/02
PB - arXiv
DO - 10.48550/arxiv.2609.02884
UR - https://arxiv.org/abs/2609.02884
AB - 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≤ 16 and bounded for n≥ 17. We establish that C_n is convergent whenever n≤ 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.
ER -