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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Sources and scope.

The analytic chain above is ordinary mathematics. Its general inputs are the Riemann mapping theorem, the Bergman kernel, the argument principle, the coarea formula, and Pólya’s area inequality $Area {| f | \leq t} \leq π t^{2 / n}$ [1], [2]. The exact rational certificate is checked by a separate exact-arithmetic program, not by the Lean kernel. The two additional root-count bounds above are the ordered-distance estimate in the earlier distance comparison and the area-rearrangement estimate in the hyperbolic packing argument. Their derivations above are part of the analytic proof. No independent review of this theorem is recorded. Prior art for the assembled statement is unassessed. No novelty is claimed for the slice inequality, which follows directly from disjointness of the balls and the hyperbolic law of cosines. A research note carries the full analytic argument, the table of certified weighted bounds and the instructions for running the certificate. The stopping-time test requires $μ < e^{- X}$ ; the stated $13 / 25$ theorem and the $529 / 1000$ consequence are two rational cutoffs within that range. Other theorems apply to some polynomials outside it, so failure of this numerical test does not identify the class left untreated by all the results. Fixed-degree arguments remain stronger at $n = 4$ and $n = 5$ , where the corresponding thresholds are $61 / 100$ and $139 / 250$ ; from $n = 6$ on the all-degree constant $13 / 25$ is the better statement. These threshold arguments do not settle the unrestricted existence of a short curve joining two roots.

Remark 4.5 (unformalised scaled path bound). Every squarefree monic polynomial $f$ of degree $n \geq 2$ has two distinct zeros joined by a rectifiable curve of length less than $(5 / 2) μ^{1 / n}$ in ${| f | < (25 / 13) μ}$ , where $μ = min_{f^{'} (c) = 0} | f (c) |$ .

Proof. For $n \geq 3$ , apply Remark 4.1 to $g (z) = s^{- n} f (s z)$ with $s = ((25 / 13) μ)^{1 / n}$ . Its least critical modulus is $13 / 25$ ; rescaling gives length less than $2 (25 / 13)^{1 / n} μ^{1 / n} < (5 / 2) μ^{1 / n}$ , since $25 / 13 < (5 / 4)^{3}$ . For $n = 2$ , write $f (z) = (z - h)^{2} - d^{2}$ . The two radial segments through $h$ have total length $2 | d | = 2 μ^{1 / 2}$ and lie in the closed level $μ$ , which is inside the stated open level. ◻

The next construction retains the root count and capacity of a chosen component, at the cost of a larger universal constant.

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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