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

A two-sheeted component

The quantity $δ = 1 - | p (c) |$ measures the remaining length of the slit in the value plane. The local quadratic coefficient converts it into the spatial scale $\sqrt{δ / M}$ . The radius $h$ records where that quadratic comparison is valid. Degree two will turn the local preimage estimate into a statement about the whole component.

Lemma 2.1 (a bottleneck estimate). Let $p$ be a polynomial and $U$ a component of ${| p | < 1}$ on which $p$ is a proper map of degree two. Suppose its only critical point $c$ is simple and $v = p (c) \neq 0$ . Write

p (c + z) - v = z^{2} A (z), M = | A (0) |, δ = 1 - | v | .

Suppose $h > 0$ , $| A (z) / A (0) - 1 | \leq 1 / 4$ for $| z | \leq h$ , and $δ < M h^{2} / 4$ . If $a, b$ are the two zeros in $U$ , every connected $K \subset U$ containing $a, b$ satisfies

H^{1} (K) \geq | a - c | + | b - c | - \frac{8}{3} \sqrt{δ / M} .

Proof. Cut the value disc along $J = {t v : 1 \leq t < 1 / | v |}$ . Its complement is simply connected and contains $0$ . The restriction of $p$ above this complement is an unramified two-sheeted covering, hence is the disjoint union of two biholomorphic sheets $U_{1}, U_{2}$ , containing $a, b$ respectively.

Put $r_{0} = (4 / 3) \sqrt{δ / M} < 2 h / 3$ . On $| z | = r_{0}$ ,

| p (c + z) - v | \geq \frac{3}{4} M r_{0}^{2} = \frac{4}{3} δ > δ .

For each $w \in J$ , Rouché’s theorem therefore gives two zeros, counted with multiplicity, of $p (c + z) - w$ in $| z | < r_{0}$ . They remain in $U$ : as $w$ moves out from $v$ along $J$ , the zeros continue from $c$ and cannot cross $| z | = r_{0}$ . Since the degree on $U$ is two, these are all its preimages of $w$ . Consequently $p^{- 1} (J) \cap U \subset B (c, r_{0})$ .

For $r_{0} < t < | a - c |$ , the set $U_{1} \cap {| z - c | > t}$ has relative boundary in $U$ contained in the circle $| z - c | = t$ . If that circle missed $K \cap U_{1}$ , then $K \cap U_{1} \cap {| z - c | > t}$ would be both open and closed in $K$ . It contains $a$ and excludes $b$ , contradicting connectedness. Thus the radial map $z \mapsto | z - c |$ sends $K \cap U_{1}$ onto a set containing $(r_{0}, | a - c |)$ whenever that interval is nonempty. This map is $1$ -Lipschitz, so

H^{1} (K \cap U_{1}) \geq (| a - c | - r_{0})_{+} .

The same argument applies to $b$ and $U_{2}$ . Although the two radial intervals can overlap on the real line, the sets $K \cap U_{1}$ and $K \cap U_{2}$ lie in disjoint open subsets of the plane. Hausdorff outer measure adds across that separation, even when $K$ is not assumed measurable. Consequently

\begin{aligned} H^{1} (K) & \geq H^{1} (K \cap U_{1}) + H^{1} (K \cap U_{2}) \\ \geq (| a - c | - r_{0})_{+} + (| b - c | - r_{0})_{+} \\ \geq | a - c | + | b - c | - 2 r_{0} . \end{aligned}

Substituting the value of $r_{0}$ proves the bound. Connectedness supplied the intervals; no path or parametrisation of $K$ was required. ◻

The diagram in the short paper distinguishes the value-plane slit from the two radial projections used above.

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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