Plectis

Long paper

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

Erdős #1041 By 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

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The short paper develops the counterexample and the trinomial argument. Here Section 2 supplies the exact tests for the polynomial, while Sections 6 and 9 give further explicit paths. The area argument in Section 4 and the isolated-value construction in Section 7 are ordinary analytic proofs, still requiring formalisation. Their conclusions appear as remarks. The local notes identify the assumptions of the existing Lean deductions; Section C collects the formal and computational references.

The question and the degree-seven counterexample

The degree-seven construction

Trinomials

A small least critical value

A path estimate from area and boundary length

Degree three

An isolated simple critical value

Examples separating the geometric hypotheses

Collinear roots and two sparse polynomial families

Further families and counterexamples to proposed proof steps

The Newton value equation

Separating the rays of critical values

Why the proposed spanning-tree estimate fails

Numerical path searches

Further questions about topology, length and perturbation

Numerical estimates for the packing inequality

Guide to the results

Formal statements and computational checks

References

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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