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

The question and the degree-seven counterexample

For a monic polynomial $f$ , let $Ω_{f} = {| f | < 1}$ ; the notation $E_{f}$ used in the segment arguments denotes the same open set. We write $K_{t} (f) = {| f | \leq t}$ for a closed sublevel. Erdős, Herzog and Piranian [4] asked:

Problem 1.1 (Erdős #1041). Let $f (z) = \prod_{i = 1}^{n} (z - z_{i})$ be monic of degree $n \geq 2$ , with $z_{i} \in D$ , the open unit disc. Must two of the roots be joined by a curve of length less than $2$ lying in the open lemniscate ${z \in C : | f (z) | < 1}$ ?

Boundary roots already give an elementary failure of strict containment. For $f (z) = z^{2} - 1$ , every continuous path from $- 1$ to $1$ meets the imaginary axis, where $| f (i y) | = 1 + y^{2} \geq 1$ . The two endpoints are therefore unjoinable in ${| f | < 1}$ , although the diameter lies in ${| f | \leq 1}$ . This illustrates the distinction between the open and closed lemniscates.

A repeated zero gives a constant path between two occurrences. Thus the question concerns distinct zeros. The catalogue of Bloom [3] records the problem and its discussion. Pendyala [11] proved the degree-four case by chords or radial segments through the centre of a smallest enclosing disc. The degree-seven construction of ani has a different geometry.

Theorem 1.2 (ani’s counterexample). The monic polynomial in (2) has seven distinct roots in the open unit disc. Every connected subset of its strict unit lemniscate containing two roots has one-dimensional Hausdorff measure greater than $2$ .

Five components contain one zero each; the remaining component contains two, and the polynomial has degree two on it. A quadratic estimate near its critical point counts two inverse images, with multiplicity, for each value on an outward slit. Since the component has degree two, these are all its inverse images of that value. Hence the entire preimage of the slit in the component lies in the small disc where the local estimate applies.

On each of the two inverse sheets, radial projection gives a lower bound for the Hausdorff measure of a connected set joining the zeros. The sheets are disjoint, so the two bounds can be added. Their sum exceeds $2$ . Section 2 supplies the estimates and proof. Applied to a continuous path image, this also excludes a rectifiable path of total variation less than $2$ .

The theorem concerns a fixed polynomial. The small-parameter family reported by ani is not proved here, and independent human review of correspondence with the 1958 wording is not recorded. The supplied formal statements and their scope are listed in Section C.

For trinomials the root identity keeps every root-to-origin segment inside the lemniscate, giving a short broken line between any pair of zeros. The later analytic remarks seek a pair by following sublevel components: one argument uses a small least critical-value modulus, and another uses an isolated simple critical value. Their conclusions have additional formalisation requirements. A bound on critical-value means alone makes no choice of a contained path.

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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