From $\log 2$ to $\pi/2$: the sharp asymptotic inradius of polynomial lemniscates

YuJian Geng; Dong Qiu

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arXiv · 2609.06461

From $\log 2$ to $\pi/2$: the sharp asymptotic inradius of polynomial lemniscates

YuJian Geng·Dong Qiu

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Abstract

Let $R_n$ be the infimum of the inradii of $\{z:|p(z)|<1\}$ over monic degree-$n$ polynomials whose zeros lie in the closed unit disk. We prove $nR_n\to\pi/2$, matching the asymptotic obstruction supplied by $z^n-1$. We first establish the exact universal radius $2^{1/n}-1$ for disks centered at zeros, which recovers the $(\log2)/n$ bound. Small inradius then forces radial concentration of the zeros and decay of their low reciprocal moments. These estimates give an entire limit with a modulus reflection identity; a second rescaling produces an exponential tangent and strict sublevel disks of every radius below $\pi/2$. The proof is accompanied by a Lean~4 formalization and a step-by-step source index.

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BibTeX RIS

YuJian Geng, Dong Qiu. 2026-09-06. From $\log 2$ to $\pi/2$: the sharp asymptotic inradius of polynomial lemniscates. https://arxiv.org/abs/2609.06461

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