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Claim. The preprint of V. S. Pendyala, Radial Access for Polynomial Lemniscates: The Cubic Theorem and a Degree-Six Obstruction (SSRN, doi:10.2139/ssrn.6850818), asks when the component of $E=\{z:\lvert f(z)\rvert\leq 1\}$ containing $0$ contains a full radius from $0$ to the unit circle, for monic $f$ with all zeros in the closed unit disc. Its abstract asserts two results. Every such $f$ of degree at most three has such a radius. Some such $f$ of degree six, with all zeros in the open disc, blocks every radius; a finite rational certificate checked in exact arithmetic verifies the example. A path from $0$ to the unit circle has length at least $1$, with equality only along a radius, so for Problem 1120 the first result gives $S(n)=1$ for $n\leq 3$, and the second gives a degree-six polynomial whose shortest escape is longer than $1$, so $S(6)>1$. The first degree with a blocked radius therefore lies between $4$ and $6$; the preprint leaves its exact value open.

Submission note. Posted to the site's forum by Venkata Siddharth Pendyala on 18 June 2026:

I have proved Erdős’s conjecture for this problem that the extremal shortest path length tends to infinity with $n$, but not too fast. More precisely, if $S(n)$ denotes the largest possible shortest length of a path in

$$> E_f={z\in\mathbb C: |z|\le 1,\ |f(z)|\le 1} >$$

joining $0$ to $\partial\mathbb D$, over all monic degree-$n$ polynomials whose zeros lie in $\overline{\mathbb D}$, then for all sufficiently large $n$,

$$> c\sqrt{\log n}\le S(n)\le \pi n >$$

for an absolute constant $c>0$. In particular, $S(n)\to\infty$, while the sharp asymptotic order remains open. The paper for this is available as an arXiv preprint at: https://arxiv.org/abs/2606.19178

A secondary paper I have written studies the length-one extreme of Erdős’s lemniscate path problem. While the main paper asks how long the shortest escape path in $E_f$ can be, this note asks when the optimal path can be a straight radius. I prove that every admissible polynomial of degree at most $3$ has such radial access, but give a certified degree-$6$ example for which every radius is blocked. Thus the first degree where Erdős’s path problem can fail to have a length-one extremal escape lies between $4$ and $6$.

This secondary paper is available as an SSRN preprint at: https://dx.doi.org/10.2139/ssrn.6850818

Covers. The value $S(n)=1$ for $n\leq 3$, where $S(n)$ is the worst-case shortest escape length defined on the claim page of the bounds, and the strict inequality $S(6)>1$. It says nothing about the growth of $S(n)$.

Standing. A single-author preprint, not refereed and with no outside review, announced in the problem's thread on 2026-06-18 together with the author's arXiv preprint on the bounds. The site labels the problem OPEN.

Depends on. No page of this wiki.