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 lower bound for the number of roots.

Equation (8) already gives $k \geq 2 x / δ (a)$ : the required sum is at least $x$ , and each root contributes at most $δ (a) / 2$ . Improving this estimate requires the pairwise separation, not just the distance of each root from the origin. Testing only consecutive pairs in angular order loses essential information. For $k = 2 m$ , put $m$ points at hyperbolic radius $d_{low}$ and $m$ at radius $d_{low} + D$ , alternating in cyclic order. The reverse triangle inequality separates every consecutive pair by at least $D$ , regardless of the angular gaps, whereas

\sum_{j} λ (d_{j}) = m (λ (d_{low}) + λ (d_{low} + D)) ⟶ \infty (m \to \infty) .

The even number of points is needed for cyclic alternation. These point configurations test only the consecutive-pair constraints; they need not satisfy separation for nonconsecutive pairs or arise from a polynomial component. To use the missing pairwise information, we impose the packing bound on every circle centred at the origin.

Lemma 4.3 (circle-slice packing). Under (7), for every $r > 0$ ,

\sum_{j = 1}^{k} w (d_{j}, r) \leq π, w (d, r) = \arccos (clamp \frac{\cosh d \cosh r - \cosh (D / 2)}{\sinh d \sinh r}),

where $clamp$ truncates its argument to $[- 1, 1]$ .

Proof. The open balls $B_{j} = B_{hyp} (b_{j}, D / 2)$ are pairwise disjoint, since a common point would force $d (b_{i}, b_{j}) < D$ . Fix $r > 0$ . By the hyperbolic law of cosines the point of the hyperbolic circle of radius $r$ at angle $θ$ lies in $B_{j}$ exactly when $\cosh d_{j} \cosh r - \sinh d_{j} \sinh r \cos (θ - θ_{j}) < \cosh (D / 2)$ . The set of angles satisfying this inequality has measure $2 w (d_{j}, r)$ : clamping gives measure zero for an empty intersection and $2 π$ for a full circle, with tangent endpoint sets of measure zero. The intersections of the balls with this circle are disjoint, so their angular measures sum to at most $2 π$ . ◻

Remark 4.4 (unformalised application of a bound for the number of roots). Fix radii $r_{1}, \dots, r_{p} > 0$ and weights $σ_{1}, \dots, σ_{p} \geq 0$ , put $Σ = \sum_{i} σ_{i}$ and

U = sup_{d \geq d_{low} (a)} [λ (d) - \sum_{i} σ_{i} w (d, r_{i})], λ (d_{low} (a)) = \frac{δ (a)}{2} .

If $U > 0$ , then failure forces $k \geq (x - π Σ) / U$ .

Proof. By (8), Lemma 4.3 and the radius bound,

x \leq \sum_{j} λ (d_{j}) = \sum_{j} [λ (d_{j}) - \sum_{i} σ_{i} w (d_{j}, r_{i})] + \sum_{i} σ_{i} \sum_{j} w (d_{j}, r_{i}) \leq k U + π Σ . ◻

◻

A nonnegative weighted sum of the circle-slice inequalities bounds every admissible configuration at once. The weights give a usable bound only when $Σ$ and the supremum $U$ have rigorous bounds. In particular, checking the expression on a finite grid does not bound the supremum between grid points. Zero weights recover only the individual bound $k \geq 2 x / δ (a)$ ; nonzero weights incorporate the circle-packing information and can improve the result. A certificate at area $a^{'} \geq a$ also applies at $a$ . Indeed, $d_{low} (a) \geq d_{low} (a^{'})$ , so the admissible range of $d$ only shrinks. Also $D (a) \geq D (a^{'})$ , which enlarges every slice angle $w (d, r_{i})$ at fixed $d, r_{i}$ . Since the weights are nonnegative, the expression defining $U$ can only decrease. Thus the same certified pair $(Σ, U)$ remains valid. The numerical comparison also uses two bounds that do not involve chosen weights. They are useful where the circle-slice certificate is weaker. We derive them here so that every root-count input to the comparison is visible.

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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