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arXiv · 2606.19178

Shortest paths in polynomial lemniscate sublevel sets and a problem of Erdős

Venkata Siddharth Pendyala
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Abstract

Let $f(z)=\prod_{j=1}^{n}(z-a_j)$ be monic, with all zeros in the closed unit disk, and put $E_f=\{z\in\mathbb{C}: |z|\leq 1,\ |f(z)|\leq 1\}$. Let $S(n)$ be the largest possible shortest length of a path in $E_f$ joining $0$ to $\partial\mathbb{D}$, where the maximum is taken over all such polynomials of degree $n$. We prove that, for all sufficiently large $n$, $c\sqrt{\log n}\leq S(n)\leq πn$ with an absolute constant $c>0$. This proves the qualitative unboundedness predicted by Erdős. The proof combines an explicit geometric maze, Green-function and Faber-polynomial estimates, analytic quantization of circle measures, and a reciprocal-sweeping upper bound.

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Venkata Siddharth Pendyala. 2026-06-17. Shortest paths in polynomial lemniscate sublevel sets and a problem of Erdős. https://arxiv.org/abs/2606.19178

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