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

Examples separating the geometric hypotheses

The two critical-value hypotheses leave different restrictions on a polynomial. We first give a cubic satisfying neither of them. A second cubic shows why the root count at the first merger cannot determine capacity at a later level.

Proposition 8.1 (failure of two critical-value criteria to cover all polynomials). Let $f (z) = z^{3} + (3 / 100) z - 3 / 4$ . Every root lies in the open unit disc, both critical points are simple, the critical values lie on distinct positive rays, $μ > 13 / 25$ , and

| 1 - f (c_{-}) / f (c_{+}) | < 2 / 375 < 2.

Proof. On $| z | = 1$ , $| 3 z / 100 - 3 / 4 | \leq 78 / 100 < 1$ , so Rouché’s theorem puts all three roots in the disc. The critical points are $c_{\pm} = \pm i / 10$ , and $f (c_{\pm}) = - 3 / 4 \pm i / 500$ . They are nonzero, have distinct arguments and modulus greater than $3 / 4$ . Moreover

| 1 - \frac{f (c_{-})}{f (c_{+})} | = \frac{1 / 250}{| f (c_{+}) |} < \frac{2}{375} .

For any $w_{0} \in [0, 1]$ , the other normalised value is at distance less than $1 + 2 / 375 < 4 / 3$ from $w_{0}$ , so the radius- $4 / 3$ hypothesis fails. ◻

A component with two roots immediately after the first merger may contain more roots at a later level. Thus its first root count alone does not give a capacity saving at level $2 μ$ .

Proposition 8.2 (root count does not force a capacity gap). Let $g (z) = z^{3} - (3 / 400) z - 3 / 32$ . All roots lie in the open unit disc and $μ = 187 / 2000 \leq 1 / 2$ . The first merger joins two root components, so $k_{0} = 2$ , but the component at level $2 μ$ containing that pair has normalised capacity $1$ .

Proof. Rouché’s theorem on the unit circle applies because $3 / 400 + 3 / 32 < 1$ . The critical points are $\pm 1 / 20$ , with critical values $- 187 / 2000$ and $- 188 / 2000$ . The first merger therefore has one simple critical point and joins two root components. At level $2 μ = 374 / 2000$ , both critical points are inside the filled lemniscate. The component count $n - (n - 1) = 1$ from Riemann–Hurwitz shows that it is connected. Its capacity is $(2 μ)^{1 / 3}$ [28], so division by that scale gives one. The use of logarithmic capacity agrees with transfinite diameter by the Fekete–Szegő theorem [28]. ◻

The degree-eight perimeter example, Proposition 5.5, requires no root-disc hypothesis. It refutes the proposed Gamma constant, while the existence of a uniform one-root perimeter bound remains open in Problem 5.4. The Runge construction reported in [26] has no supplied proof and is not used.

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.

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