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Claim. Call a monic polynomial $P$ of degree $n$ non-degenerate when every critical value has modulus at least $1$, so that no branch point lies over the open unit disk and every inverse branch at a root extends univalently across it. If $P$ is non-degenerate with all roots in the closed unit disk, then $\rho(P)\ge1/(4n)$, where $\rho$ is the inradius of $\{z:\lvert P(z)\rvert<1\}$ as in Problem 1039. The argument, posted on the site's discussion thread on 2026-03-06 by the user Houi with Claude Opus 4.6 named as the system used: the Koebe quarter theorem puts a disk of radius $1/(4\lvert P'(z_j)\rvert)$ about each root inside the lemniscate, and the Fekete energy of $n$ points in the closed disk, maximized by the roots of unity, gives $\sum_j\log\lvert P'(z_j)\rvert\le n\log n$, so some root has $\lvert P'(z_j)\rvert\le n$. The manuscript also reduces the general question to the non-degeneracy of an inradius minimizer and describes an obstruction to the conformal approach in the degenerate case through Blaschke products.

Covers. Polynomials whose critical values all have modulus at least $1$, with the bound $\rho(P)\ge1/(4n)$. It says nothing about polynomials with a critical point inside the lemniscate, which is the case the general question turns on.

Depends on. No page of this wiki.

Standing. Claimed. Nat Sothanaphan's AI check of the first version claimed several gaps or omissions; the author revised the manuscript several times, and a reader found a gap in Lemma 5.1 and an example that violated the squarefree hypothesis, both of which the author replaced. The repository's Lean file, produced with the Aristotle system, proves the main theorem from axiomatized classical inputs (the Koebe quarter theorem, Hadamard's inequality, existence of univalent branches, existence of a minimizer, and two statements needing the Riemann mapping theorem); Sothanaphan's AI check read it as a conditional proof of the partial result. Erik Lundberg wrote on the thread (2026-06-24) that the same combination of the Koebe theorem and Hadamard's inequality for a Vandermonde determinant is in the proof of Proposition 18(b) of Krishnapur, Lundberg and Ramachandran [KLR25], which produces an order $1/n$ inradius bound under an equivalent hypothesis, so the result may not be new. No proof claim was registered on the site's proof-claims tab and no refereed or arXiv version exists. The general lower bound, without the non-degeneracy hypothesis, is the later claim on Price 2026.