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

Will Cook

Long paper

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

Erdős #1041 By Will Cook 29 September 2026 Authorship, AI use and citation

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Summary. The record gives the rational estimates and geometric proof for ani's degree-seven polynomial: every connected set in its strict unit lemniscate joining two roots has Hausdorff measure greater than two. It develops contained paths for trinomials, collinear roots and two sparse families. Ordinary analytic arguments for critical-value criteria retain their unformalised inputs beside the statements. Critical-value moments, counterexamples to prescribed paths and questions about flow-strip geometry complete the account. The one-polynomial formal statement and correspondence with the historical wording remain distinct.

This paper covers the exact geometry of ani's one-polynomial counterexample, proved positive families and critical-value arguments with their unformalised inputs identified; formal-statement status and historical correspondence remain owned by the claims registry.

It does not establish the validity of claims tagged Lean, which belongs to the cited kernel-checked source, or the current status of Erdős #1041, including the later one-polynomial Formal Conjectures path-image Hausdorff-length check and its unreviewed historical curve-length correspondence; those distinctions belong to docs/claims.json in the mathematics repository.

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In this paper

The degree-seven construction

The construction has two separate quantitative requirements. The two roots in the selected component must have combined distance from its critical point greater than $2$ , and the preimage of an outward value slit must be small enough to preserve that surplus. A narrow neck alone would not suffice: for $p (z) = z^{2} - r^{2}$ , $0 < r < 1$ , the slit $(- 1, - r^{2}]$ has preimage ${i y : | y | < \sqrt{1 - r^{2}}}$ , while the root segment $[- r, r]$ still has length $2 r < 2$ and lies in ${| p | < 1}$ .

We first prove the estimate that separates these requirements. We then verify root locations, critical values and the membership of the chosen pair in one component. The final fixed-parameter estimates compare the root-distance surplus with the neck loss, and distinguish both from the radial contraction $1 - ρ = ε^{8}$ .

A two-sheeted component

Polynomial and roots

Critical points and local estimates

The two roots in the component

The length inequality

About this paper

Authorship and AI use. Will Cook built and directed the research infrastructure and maintains the public release. He reviewed claims when he could. AI agents did most of the research and drafting. Cook did not independently verify every claim.

Cite and contact. Cite this paper by its title, author and date above, with its PDF; cite the earlier sources it uses for a mathematical result. For software, use the release citation and give the commit used. Contact Will with questions or corrections.

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