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

PDFOpen the PDF, 86 pages TeXLaTeX source Source on GitHub Problem page

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.

Start here, by the editorial route

In this paper

A sufficient critical-value threshold in degree five

The reciprocal balance at a critical point bounds the sum of the two nearest root distances. We begin with this formal geometric statement, then distinguish distance from containment in the degree-five argument.

Theorem 10.5 (a geometric-mean bound for distances to a critical point). Let $n \geq 2$ , let $z_{1}, \dots, z_{n}, c \in C$ with $c \neq z_{k}$ for every $k$ , and suppose

\sum_{k = 1}^{n} \frac{1}{c - z_{k}} = 0.

If $r > 0$ is determined by

r^{n} = \prod_{k = 1}^{n} | c - z_{k} |,

then there are distinct indices $i, j$ such that

| c - z_{i} | + | c - z_{j} | \leq 2 r .

Proof. Relabel the points so that their distances satisfy $d_{1} \leq d_{2} \leq \dots \leq d_{n}$ . The reciprocal balance and the triangle inequality give

\frac{1}{d_{1}} = | \sum_{j = 2}^{n} \frac{1}{c - z_{j}} | \leq \sum_{j = 2}^{n} \frac{1}{d_{j}} \leq \frac{n - 1}{d_{2}} .

Thus $d_{2} \leq (n - 1) d_{1}$ . Also $d_{1} d_{2}^{n - 1} \leq r^{n}$ , since every remaining distance is at least $d_{2}$ . For $t = d_{2} / d_{1} \in [1, n - 1]$ ,

\frac{d}{d t} \log \frac{2 t^{(n - 1) / n}}{1 + t} = \frac{n - 1 - t}{n t (1 + t)} \geq 0.

The logarithm vanishes at $t = 1$ , so $d_{1} + d_{2} \leq 2 (d_{1} d_{2}^{n - 1})^{1 / n} \leq 2 r$ . ◻

The distance bound for the two nearest roots.

Two failures of straight segments.

Containment of the two selected segments.

Area of a connected sublevel set.

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.

Prefer the manuscript?

PDFOpen the PDF, 86 pages TeXOpen the LaTeX source Problem page

Equations typeset from the exact TeX. Rendered from the LaTeX manuscript source with file fingerprint sha256:4bee9a881dbbe940. Its file fingerprint matched the published source record, so the PDF above, the LaTeX, and this page are one manuscript. Redeploying the site regenerates this page from whatever the public repository holds at that moment.