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Claim. Let $f(z)=z^n+az^m+b$ with $1\le m<n$ be monic with every zero in the open unit disk. Every root $\zeta$ is joined to the origin by the segment $\{t\zeta:0\le t\le1\}$ inside $\{z:\lvert f(z)\rvert<1\}$, because the root equation gives $f(t\zeta)=b(1-t^m)+\zeta^n(t^n-t^m)$ and $\lvert b\rvert<1$ by Vieta, so $\lvert f(t\zeta)\rvert<1-t^n\le1$ for $0\le t<1$; two distinct roots are therefore joined through the origin by a path of length $\lvert\zeta_1\rvert+\lvert\zeta_2\rvert<2$, which is the question of Problem 1041 for this family. The same note proves the critical-value estimate $\sum_{j<n}\lvert f(c_j)\rvert^{2/(n-1)}\le n-1$ for any monic $f$ of degree $n\ge2$ with zeros in the closed unit disk, sharp for $z^n-\lambda$ with $\lvert\lambda\rvert=1$, and that a squarefree monic polynomial of degree at least two whose least critical value has modulus at most $13/25$ has two distinct roots joined by a curve of length less than $2$ in the lemniscate, without a root-location hypothesis. William Cook posted the notes on the site's discussion thread on 2026-09-11 as one of eight problems in Cook's Plectis repository, opening with the statement that Cook had Astra write up the positive family, and stating that AI tools contributed substantially under their direction and that the notes have had no independent review. The Abel identity, the trinomial radial inequalities and the length bound have public Lean source at a cited revision; the critical-value and low-critical-value arguments are ordinary proofs, not Lean theorems.

Covers. Trinomials $z^n+az^m+b$ with all zeros in the open unit disk, in every degree, and squarefree polynomials with a critical value of modulus at most $13/25$. It is not a general path theorem, as the author says, and the general question is claimed false in degree seven on ani 2026 (counterexample), which the same author checked and formalized.

Depends on. No page of this wiki.

Standing. Claimed. The thread records no check of the notes, no proof claim was registered on the site's proof-claims tab, and no refereed or arXiv version exists. The author asked whether the critical-value estimate has an earlier reference.