Let p∈C[z] be a centered polynomial of degree at least 2, let Z(p) denote its set of roots, and let C_p be the Chebyshev map C_p(z)=z-fracp(z)p'(z)-fracp''(z)p(z)^22p'(z)^3. Define Sigma_p=sigma∈operatornameAff(C):sigma(Z(p))=Z(p) and Sigma_C_p=sigma∈operatornameAff(C):sigma(J(C_p))=J(C_p), where J(C_p) is the Julia set of C_p. The Nayak–Pal conjecture asserts that Sigma_p=Sigma_C_p for every such polynomial p.

Nayak–Pal symmetry conjecture for Chebyshev maps

Let $p\in\mathbb{C}[z]$ be a centered polynomial of degree at least $2$ , let $Z(p)$ denote its set of roots, and let $C_p$ be the Chebyshev map $C_p(z)=z-\frac{p(z)}{p'(z)}-\frac{p''(z)p(z)^2}{2p'(z)^3}$ . Define $\Sigma_p=\{\sigma\in\operatorname{Aff}(\mathbb{C}):\sigma(Z(p))=Z(p)\}$ and $\Sigma_{C_p}=\{\sigma\in\operatorname{Aff}(\mathbb{C}):\sigma(J(C_p))=J(C_p)\}$ , where $J(C_p)$ is the Julia set of $C_p$ . The Nayak–Pal conjecture asserts that $\Sigma_p=\Sigma_{C_p}$ for every such polynomial $p$ .

References

Primary source

arXiv

Additional references

Chebyshev's method applied to polynomials with rotational symmetry — arXiv

complex-dynamics holomorphic-dynamics

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed progress

A new unrefereed manuscript claims progress on the conjecture for an infinite rotationally symmetric family, while the general conjecture remains open.

Nayak and Pal proposed that a centered polynomial’s symmetries coincide with those of the Julia set of its Chebyshev map. Their August 24, 2022 preprint labels this equality as Conjecture 1 and proves only special cases.

Known results

For every centered polynomial, $\Sigma_p\subseteq\Sigma_{C_p}$ (Nayak and Pal, 2022).
Equality holds for unicritical polynomials and for polynomials with exactly two equally multiple roots (Nayak and Pal, 2022).
Equality holds for specified cubic and quartic cases; the corresponding Julia sets are connected and locally connected, with Fatou set given by root basins (Nayak and Pal, 2022).

September 2, 2026 claimed family result

A manuscript claims to prove the conjecture for $p_n(z)=z(z^n-1)$ , while also establishing connectivity and basin properties of the associated Julia sets. This extends the recorded special cases but does not, on the available evidence, settle the general conjecture; the manuscript is unrefereed.

Current status (as of September 2026): The equality $\Sigma_p=\Sigma_{C_p}$ is claimed for the family $p_n(z)=z(z^n-1)$ , but the general conjecture remains open and the new claim is unverified.

Sources

No solutions have been posted yet.