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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Degree three

For a cubic we can estimate the inverse roots directly after normalising at a critical point of least value modulus. The root-disc hypothesis makes that modulus less than one, and two radial segments in the normalised coordinates give a path shorter than $2$ .

Theorem 6.1 (the cubic case). Let $f (z) = \prod_{j = 1}^{3} (z - z_{j})$ with $| z_{j} | < 1$ , the roots listed with multiplicity. Then two listed root occurrences are joined inside ${| f | < 1}$ by a polygonal path of length strictly below $2$ . If $f$ is squarefree the two are distinct.

Proof. Multiple roots give a constant path, so assume $f$ squarefree. The Erdős–Herzog–Piranian component lemma [4] gives a component of ${| f | < 1}$ containing two roots. Join them by a compact path in that open component, and choose a regular value modulus $t < 1$ larger than $max | f |$ on the path. The component of ${| f | < t}$ containing it has $k \geq 2$ roots and hence $k - 1 \geq 1$ critical points, by the component-wise Riemann–Hurwitz count in the proof of [8]. One of these critical points has value of modulus below $t < 1$ . This choice of $t$ avoids assuming that the level $| f | = 1$ is regular.

Suppose first that $f^{'}$ has two distinct zeros. Choose a critical point $c$ minimising $| f (c) |$ , write the other one as $c + δ$ , and put $v = f (c)$ , so $0 < | v | < 1$ . Monicity gives the exact expansion $f (c + d) = d^{3} - \frac{3}{2} δ d^{2} + v$ . Choose $α$ with $α^{3} = v$ and set $b = δ / α$ ; dividing by $v$ gives $f (c + α w) / v = P_{b} (w)$ with

P_{b} (w) = w^{3} - \frac{3}{2} b w^{2} + 1 .

At the other critical point $f (c + δ) = v (1 - b^{3} / 2)$ , so minimality of $| v |$ is exactly the hypothesis $| 1 - b^{3} / 2 | \geq 1$ .

Under that hypothesis $P_{b}$ has at least two zeros, with multiplicity, in the closed unit disc. In the strict region a unit-circle zero $w$ would give $b = \frac{2}{3} (w + w^{- 2})$ . Write $w^{3} = e^{2 i u}$ ; substitution gives

| 1 - b^{3} / 2 |^{2} = 1 + \frac{64}{729} \cos^{4} u (16 \cos^{2} u - 27) \leq 1.

Since $16 \cos^{2} u - 27 < 0$ , equality requires $\cos u = 0$ , which gives $b = 0$ . Thus the strict region has no unit-circle zero. The radial deformation $b \mapsto s b$ , $s \geq 1$ , stays in the strict region, since $z = b^{3} / 2$ and $| 1 - z |^{2} > 1$ give $| 1 - s^{3} z |^{2} - 1 = s^{3} (s^{3} | z |^{2} - 2 ℜ z) > 0$ . For large $s$ , Rouché’s theorem on $| w | = 1$ compares $P_{s b}$ with $- \frac{3}{2} s b w^{2}$ , whose modulus $\frac{3}{2} s | b |$ exceeds $2 \geq | w^{3} + 1 |$ , so $P_{s b}$ has exactly two zeros in the open unit disc; no zero crosses the circle along the deformation. For the equality case with $b \neq 0$ , the same radial deformation lies in the strict region for every $s > 1$ , and continuity of the root multiset allows $s ↓ 1$ . If $b = 0$ , then $P_{b} (w) = w^{3} + 1$ and all three roots lie on the unit circle.

If $P_{b} (w) = 0$ and $0 \leq t \leq 1$ , then $P_{b} (t w) = 1 - t^{2} - t^{2} (1 - t) w^{3}$ , so $| w | \leq 1$ gives $| P_{b} (t w) | \leq (1 - t^{2}) + t^{2} (1 - t) | w |^{3} \leq 1 - t^{3} \leq 1$ . The whole segment from $0$ to $w$ therefore lies in the closed unit sublevel set. Selecting two normalized roots $w_{1}, w_{2}$ with $| w_{i} | \leq 1$ , the two segments from $c$ to $c + α w_{i}$ lie in ${| f | \leq | v |} \subset {| f | < 1}$ and have combined length at most $2 | α | = 2 | v |^{1 / 3} < 2$ .

If instead $f^{'}$ has a double zero $c$ , then $f (c + d) = d^{3} + v$ . With $α^{3} = - v$ and $ω = e^{2 π i / 3}$ , its roots are $c + α$ , $c + α ω$ , $c + α ω^{2}$ . Averaging their squared moduli gives $| c |^{2} + | α |^{2} < 1$ , so $| α | < 1$ . Any two segments from $c$ to the roots have total length $2 | α | < 2$ , and along either segment the value has modulus $| v | (1 - t^{3}) < 1$ away from the root endpoint. ◻

The unit level need not be regular, even when all roots lie in the open unit disc. For example, $Q (z) = (z - 1) (z + 1) (z + 9 / 10)$ has $Q (1 / 3) = - 148 / 135 < - 1$ . Its minimum $v$ on $[- 9 / 10, 1]$ occurs at an interior critical point and satisfies $v < - 1$ . Set $r = | v |^{- 1 / 3} < 1$ . The monic cubic $r^{3} Q (z / r)$ has the three distinct roots $- r, - 9 r / 10, r$ in the open unit disc and a critical value equal to $- 1$ . Its unit level is therefore singular, although the cubic path theorem applies.

Sources and scope.

Pendyala [11] proves the separate quartic case; no historical priority for the cubic theorem is asserted here.

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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