Paths in Polynomial Lemniscates: A Degree-Seven Counterexample and Radial Connections

Sources and scope.

The inputs of the construction in the proof of Remark 5.1 are the capacity identity for polynomial preimages [2], Pólya’s area inequality [2], Cauchy–Schwarz, the Koebe distortion theorem for univalent maps, and averaging over adjacent boundary endpoints. The public source of record is the full area and boundary-length argument, which carries every rational verification quoted above. Its Section 10 gives a weaker conditional bound involving an auxiliary parameter and a logarithm; Remark 5.6 states the stronger $\beta \rho$ consequence proved here. That section’s proposed numerical perimeter constant is refuted by the degree-eight example above and is not assumed in the theorem. The unconditional construction retains level $2\mu$ and constant $71/10$. The scaled low-critical corollary has both a smaller constant and a smaller level. The proof retained here is ordinary, with component-sensitive consequences on their stated regions; its rational cutoffs are exact integer inequalities. The earlier perimeter samples concerned only a selected component. They do not test the stronger condition that the level lie below every critical value. The existence of a root-to-root curve of length less than $2$ is not established for the unrestricted class by the arguments in this record. The adjacent classical literature on lemniscate length concerns the arclength of the level curve $\{|p|=1\}$, which is Erdős #114. Fryntov and Nazarov recall its history [15]: Dolzhenko’s bound $4\pi n$, Pommerenke’s bound $74n^2$ of 1961, and Borwein’s bound $8\pi en$ [13]. Eremenko and Hayman proved $9.173n$ [14], Fryntov and Nazarov the asymptotically sharp $2n+o(n)$ [15], and Tao resolved the problem for large $n$ [16]. These results bound the length of a level curve, not the least internal root-pair path. This distinction identifies the quantity being estimated; no claim of novelty follows from the scope or outcome of a literature search.