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

Area of a connected sublevel set.

The following calculation applies in every degree $n\ge 2$, independently of the degree-five segment criterion. Let $f$ be monic of degree $n$ and let $t>0$ with $K_t=\{|f|\le t\}$ connected. Translate its root centroid to $0$, so $f(z)=z^n+c_{n-2}{z}^{n-2}+\cdots$, and let $\psi$ be the exterior map normalised at infinity with positive leading coefficient. For a regular connected lemniscate, the identity $f(\psi (\zeta ))=t{\zeta }^n$ is [8] after scaling the value level. The following argument also permits critical boundary points. The exterior Green functions $(\log |f|-\log t)/n$ and $\log |{\psi }^{-1}|$ agree: both vanish at the boundary and have a logarithmic pole of coefficient one at infinity. Comparison there gives the leading coefficient $t^{1/n}$ for $\psi$. The quotient $f(\psi (\zeta ))/(t{\zeta }^n)$ has constant modulus one and limit one at infinity, proving the polynomial identity. Its ${\zeta }^{n-1}$ coefficient forces the constant Laurent coefficient of $\psi$ to vanish. Thus

Grönwall’s area identity takes the form

One can obtain it directly without assuming that level $t$ is regular. For $r>1$, Green’s area formula on the analytic Jordan curve $\psi (r{e}^{i\theta })$ gives the enclosed area $\pi (t^{2/n}{r}^2-\sum _{k\ge 1}k|a_k{|}^2{r}^{-2k})$. These enclosed compact sets decrease to $K_t$ as $r\downarrow 1$; continuity of area from above and monotone convergence of the series give the identity. Comparing the coefficient of ${\zeta }^{n-2}$ in the exterior identity now gives

The nonnegative coefficient sum quantifies the loss from Pólya’s upper bound. It does not by itself control an internal path length. No analogous coefficient formula for an individual component of a disconnected sublevel set is established here. The degree-five statements and their evidence classes are recorded in the degree-five estimates and their proofs.