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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An integral for successive merger scales

Consider a component containing $k$ roots that later joins a larger component as the level of $| f |$ increases. If $r \in (0, 1]$ is the ratio of these two merger scales, one logarithmic contribution assigned to each of its roots is $k^{- 1} \log (1 / r)$ . An earlier length estimate instead involves the integral

I_{k} (r) = \int_{r}^{1} \frac{d q}{q \log ((1 + q^{2 / k}) / (1 - q^{2 / k}))} .

For nearby merger scales, $r$ is close to $1$ . The integral is then much smaller than $k^{- 1} \log (1 / r)$ . The following change of variables proves that no fixed positive multiple of this logarithmic contribution is a uniform lower bound.

Theorem 10.8 (the relation between two merger-scale integrals). Define

Φ (x) = \int_{0}^{x} \frac{d t}{\log (\coth t)} (x \geq 0) .

At $t = 0$ the integrand is understood by its continuous limiting value $0$ (equivalently, the integral is improper at that endpoint). For every integer $k \geq 1$ and $0 < r \leq 1$ , with $x = k^{- 1} \log (1 / r)$ ,

\begin{matrix} (O1) & I_{k} (r) = k Φ (x) . \end{matrix}

The function $Φ$ is increasing and strictly convex on $(0, \infty)$ , and

\begin{matrix} (O2) & \frac{Φ (x)}{x} ⟶ 0 (x ↓ 0) . \end{matrix}

Consequently, for every fixed $k \geq 1$ and every $c > 0$ , some $0 < r < 1$ satisfies

\begin{matrix} (O3) & I_{k} (r) < c \frac{1}{k} \log \frac{1}{r} . \end{matrix}

In particular, no positive universal constant bounds $I_{k} (r)$ below by that constant times $k^{- 1} \log (1 / r)$ .

Proof. Set $q = e^{- k t}$ . Then $d q / q = - k d t$ , the endpoints $q = 1, r$ become $t = 0, x$ , and

\frac{1 + e^{- 2 t}}{1 - e^{- 2 t}} = \coth t,

which proves (O1). Since $\coth t$ , and hence $\log (\coth t)$ , is strictly decreasing, the positive integrand $1 / \log (\coth t)$ is strictly increasing. This proves monotonicity and strict convexity. Moreover

0 \leq \frac{Φ (x)}{x} \leq \frac{1}{\log (\coth x)} ⟶ 0,

which gives (O2); taking $r = e^{- k x}$ gives (O3). ◻

As $r$ approaches $1$ , two successive merger scales approach each other. The integral can then be arbitrarily small relative to the logarithmic contribution, even for fixed $k$ . Any estimate along a chain of mergers must retain this nonlinear dependence. Related weighted Jensen estimates and a lemma selecting two roots with small total logarithmic contribution are given in a research note. Those inequalities do not by themselves construct pieces of a contained path that meet one another.

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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