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 small least critical value

We follow a component from its first merger as the level of $| f |$ increases. Under the assumption that it contains no path shorter than $2$ between distinct roots, conformal length bounds force the roots apart. Packing then forces area growth beyond Pólya’s upper bound. The reduction to this comparison is an ordinary analytic argument; the exact computation at its end does not formalise the reduction. We retain the resulting criterion as an unformalised remark. Throughout this section

μ = min_{f^{'} (c) = 0} | f (c) |

is the least critical-value modulus.

Remark 4.1 (unformalised small-critical-value criterion). Let $f$ be squarefree and monic of degree $n \geq 2$ with $μ \leq 13 / 25$ . Then two distinct roots of $f$ are joined inside ${| f | < 1}$ by a rectifiable curve of length strictly below $2$ . No hypothesis is placed on the locations of the roots, on the number of roots in any component, or on the capacity of any component.

For $f (z) = z^{n} - b$ with $0 < | b | < 1$ , the only critical point is $0$ and $μ = | b |$ . Thus this criterion includes $| b | \leq 13 / 25$ but excludes $13 / 25 < | b | < 1$ , although the trinomial theorem gives the required path in both ranges. Its value is that it also applies to polynomials with arbitrary coefficient patterns, including roots outside the unit disc. For example, $(z - 3)^{n} - 1 / 2$ , $n \geq 2$ , has $μ = 1 / 2$ and all its roots satisfy $| z | > 2$ . The connection is required to stay in $| f (z) | < 1$ ; the example places no restriction on its distance from the origin.

Remark 4.2 (unformalised scale-free consequence). Every squarefree monic $f$ of degree $n \geq 2$ has two distinct roots joined inside ${| f | < (25 / 13) μ}$ by a curve of length below $2 ((25 / 13) μ)^{1 / n}$ .

Proof. Apply Remark 4.1 to $s^{- n} f (s z)$ with $s = ((25 / 13) μ)^{1 / n}$ , whose least critical-value modulus is $13 / 25$ , and scale back. Squarefreeness gives $μ > 0$ , so the scaling factor is positive. ◻

The proof has two steps. Failure of the length bound first forces the roots to be far apart in the hyperbolic metric of a sublevel component. Packing those roots gives a lower bound for their number. Averaging paths between adjacent boundary points then converts that root count into an area-growth inequality. The rational certificate shows that this growth would exceed Pólya’s area bound before the value level reaches $1$ .

Consequences of having no short path.

A lower bound for the number of roots.

Two further root-count bounds.

From the root count to an area contradiction.

Certificate.

Sources and scope.

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.