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PreprintarXiv licencearXiv

From log 2 to π/2: the sharp asymptotic inradius of polynomial lemniscates

  • YuJian Geng
  • Dong Qiu
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Abstract

Let Rn 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 nRn→π/2, matching the asymptotic obstruction supplied by zn−1. We first establish the exact universal radius 21/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 π/2. The proof is accompanied by a Lean~4 formalization and a step-by-step source index.

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Cite

BibTeXRIS
BibTeX
@misc{geng2026from,
  title = {{From $\log 2$ to $π/2$: the sharp asymptotic inradius of polynomial lemniscates}},
  author = {YuJian Geng and Dong Qiu},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.06461},
  eprint = {2609.06461},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.06461},
}
RIS
TY  - UNPB
TI  - From log 2 to π/2: the sharp asymptotic inradius of polynomial lemniscates
AU  - YuJian Geng
AU  - Dong Qiu
PY  - 2026
DA  - 2026/09/06
PB  - arXiv
DO  - 10.48550/arxiv.2609.06461
UR  - https://arxiv.org/abs/2609.06461
AB  - 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→π/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 π/2. The proof is accompanied by a Lean~4 formalization and a step-by-step source index.
ER  -