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

Numerical path searches

The earlier searches discretised root configurations and tested candidate connections. Their finite scope is recorded here so that a negative search is not mistaken for a general path theorem.

A search was run over random monic polynomials with roots in the unit disc. For each sample the region $\{|f|<1\}$ was rasterised and shortest grid paths were computed between every pair of roots. The reported grid distances were

No counterexample candidate was found in these trials. The reported grid distances are below $2$, but they do not certify lengths of curves in the lemniscate. A polygon whose vertices satisfy $|f|<1$ can cross the boundary along an edge. Joining an approximate root to its grid vertex introduces another edge requiring the same check. Turning a candidate into a proof needs a continuous certificate on every edge, for instance strict positivity of the real polynomial $1-|f(z(t)){|}^2$ on $[0,1]$ for each straight edge $z(t)=u+t(v-u)$, established by exact coefficients or outward-rounded interval arithmetic with subdivision, together with exact root enclosures. No such certificate is attached to the numbers above. They are grid distances for the sampled configurations, and the apparent decrease with degree is a property of the sample, from which we draw no conjecture. Even a finite collection of proved instances would not establish the universal statement. These searches do not test ani’s degree-seven counterexample. Numerical tests of a particular path construction should also examine near-degenerate saddles, thin necks, boundary-critical configurations and almost-connected separatrices. Any paths found on a grid still require verification along their entire edges.