• I’ve just uploaded to the arXiv my preprint The maximal length of the Erdős-Herzog-Piranian lemniscate in high degree. • This paper resolves (in the asymptotic regime of sufficiently high degree) an old question about the polynomial lemniscates attached to monic polynomials of a given degree , and specifically the question of bounding the arclength of such lemniscates. • For instance, when , the lemniscate is the unit circle and the arclength is ; this in fact turns out to be the minimum possible length amongst all (connected) lemniscates, a result of Pommerenke. • However, the question of the largest lemniscate length is open. • The leading candidate for the extremizer is the polynomial whose lemniscate is quite convoluted, with an arclength that can be computed asymptotically as where is the beta function. • (The images here were generated using AlphaEvolve and Gemini.) A reasonably well-known conjecture of Erdős, Herzog, and Piranian (Erdős problem 114) asserts that this is indeed the maximizer, thus for all monic polynomials of degree .
Article Summaries:
- A new arXiv preprint resolves a long‑standing question about the maximal arclength of polynomial lemniscates for monic polynomials of high degree. The author proves that, asymptotically, the conjectured extremal polynomial-whose lemniscate is highly convoluted-indeed maximizes length, confirming the Erdős-Herzog-Piranian conjecture for sufficiently large degree. Building on Fryntov-Nazarov’s sharp upper bound, the paper iteratively refines the estimate, ultimately showing the length is asymptotically (2\pi\beta(1/2,1/2)) up to a negligible error. The work also highlights the use of AI tools (AlphaEvolve) for visualizing and guiding the optimization of lemniscates.
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