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

Sources and scope.

The two-sheeted component, its square-root uniformisation, the Bergman segment inequality, the component-capacity estimate, Pólya’s area–capacity inequality and the exhaustion are ordinary arguments. They are collected in the proof using an isolated critical value, whose capacity step is cited separately above.

The proof requires both a simple critical point and a disc centred on $[0,1]$ that isolates its normalised value. For example, the trinomial $z^n-b$ with $n>2$ and $0<|b|<1$ has a short path by Theorem 3.1, but its only critical point is not simple, so this method does not apply. The restriction is on the chosen critical point and on the distance of the other critical values from the chosen disc. The other critical points may be multiple, and their values need not be separated from one another. No sharpness is claimed for $S=4/3$, and no assertion is made that every polynomial has a suitable isolated simple value. In particular, this is not a proof of the unrestricted assertion of Erdős #1041.

A related area comparison is Dubinin’s Theorem 1. The source is [12], Theorem 1 on printed page 85 of the POMI original. It concerns a holomorphic function that gives a full $n$-fold covering of an annulus $t_1<|w|<t_2$, and, in the notation of that theorem, with $E$ its explicitly defined complementary set, it states

The full covering and the explicit complementary set are hypotheses of the theorem. Tao cited it on the Erdős Problem #1041 discussion page on 25 March 2026 for the relative scaling factor $s^{2/n}$ between the areas of two nested sublevel sets, which is the reading of the displayed inequality in which $E$ and $E\cup D$ are those two sublevel sets and the covering hypothesis holds. No step of this note uses that relative inequality. The disk-family theorem instead uses Pólya’s absolute area–capacity inequality on a component containing fewer than $n$ roots, after the component-capacity estimate supplies its strict capacity gap. The absolute sublevel inequality $\mathrm{Area}\{|P|<T\}\le \pi T^{2/n}$ remains the area input to Remark 4.1 and to Remark 5.1. Dubinin’s theorem is a neighbouring result under a full covering hypothesis and supplies no curve joining two roots.