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

Consequences of having no short path.

Assume no two distinct roots are joined inside $\{|f|<1\}$ by a curve of length below $2$. Choose a critical point at level $\mu$ and two descending inverse arcs into distinct one-root components of $\{|f|<\mu \}$; below $\mu$ each component maps conformally onto the value disc, and distinct local inverse arcs at the first critical point enter distinct components, since two inverse images of a nearby regular value in one component would contradict its degree one. Their union is a compact connected set containing two roots. Fix a level $t\in (\mu ,1)$ at which no critical value has modulus $t$, so that the level is regular. Let $C_t$ be the component of $\{|f|<t\}$ containing that set, let $k\ge 2$ be its root count, and put $x=\log (t/\mu )$ and $a=\mathrm{Area}(C_t)/\pi$; Pólya’s inequality [2] gives $a\le t^{2/n}<1$. The component is a Jordan domain. List its roots as ${\zeta }_1,\dots ,{\zeta }_k$, choose a Riemann map $\phi :\mathbb{D}\to C_t$, and write $b_j={\phi }^{-1}({\zeta }_j)$. The component-wise form of [8] is

Indeed, regularity extends $\phi$ across the Jordan boundary. The quotient of the two sides without the phase has no zero or pole on the closed disc and has boundary modulus one. The maximum principle, applied to that quotient and its reciprocal, makes it a unimodular constant. The product has degree $k$, the root count of this component, not necessarily the full degree $n$. Distances between the $b_j$ below are hyperbolic distances in $\mathbb{D}$, normalised by $d(0,s)=2\mathrm{artanh}s$ for $0\le s<1$. Intrinsic distances in $C_t$ are infima of Euclidean lengths of curves in that component.

The length estimate converts area into a bound for the image of a compact real interval $I\subset (-1,1)$. Here is the coefficient calculation underlying the Bergman-kernel argument. Write ${\phi }^{\prime }(z)=\sum _{\ell \ge 0}{\alpha }_{\ell }{z}^{\ell }$. Injectivity and the area formula give $\sum _{\ell \ge 0}|{\alpha }_{\ell }{|}^2/(\ell +1)=a$. To express length as a linear integral, choose $\omega (x)=\overline{{\phi }^{\prime }(x)}/|{\phi }^{\prime }(x)|$ on $I$; the denominator is nonzero because $\phi$ is conformal. Cauchy–Schwarz gives

Compactness of $I$ permits termwise integration of the series. Since $\sum _{\ell \ge 0}(\ell +1)(xy{)}^{\ell }=(1-xy{)}^{-2}>0$ for real $x,y\in I$, taking absolute values in the resulting double integral yields

This is precisely the positivity of the disc Bergman kernel $K(z,w)={\pi }^{-1}(1-z\bar{w}{)}^{-2}$ on the real diameter. For $I=[u,v]\subset (-1,1)$, direct integration gives

Thus $I=[0,r]$ gives $\mathrm{length}(\phi ([0,r]){)}^2\le a\log (1/(1-r^2))$. Moving two preimages to $\pm s$ and taking $I=[-s,s]$ instead gives

A pair of roots at hyperbolic distance $d$ may be placed at $\pm \tanh (d/4)$. Their connecting curve then has length less than $2$ whenever $\mathrm{artanh}({\tanh }^2(d/4))<1/a$. Under our assumption that no such curve exists, we therefore have

There is also a point $h$ in the chosen compact connected set whose intrinsic distance in $C_t$ from every root is at least $1$. Otherwise the open intrinsic balls of radius $1$ about the roots would cover that set. Two balls with distinct centres cannot meet: paths from their centres to a common point would concatenate to length below $2$. But a connected set containing two distinct roots cannot be covered by these disjoint open sets. This contradiction proves the assertion. Now choose the Riemann map with $\phi (0)=h$, and continue to write $b_j={\phi }^{-1}({\zeta }_j)$. Put $d_j=d(0,b_j)$ and $\lambda (d)=-\log \tanh (d/2)$. The one-root estimate at $h$ gives $\tanh (d_j/2)\ge \sqrt{1-e^{-1/a}}$. The centre $h$ lies on the chosen inverse arcs, so $0<|f(h)|\le \mu$; it is not a root because its intrinsic distance from every root is at least $1$. The Blaschke identity at $0$ gives $|f(h)|/t=\prod _j|b_j|$, hence $\sum _j\lambda (d_j)=\log (t/|f(h)|)\ge \log (t/\mu )=x$. Together these estimates give