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 uniform central region in every degree.

For every $m \geq 1$ , the inequality for points in a smaller disc proves

| c_{j} | \leq \sqrt{1 - e^{- 2}} (1 \leq j \leq m) ⟹ \sum_{j = 1}^{m} {(\prod_{k = 1}^{m} | 1 - {\overset{―}{c}}_{j} c_{k} |)}^{1 / m} \leq m .

To see the mechanism, set $h_{j} = m^{- 1} \sum_{k} \log | 1 - {\overset{―}{c}}_{j} c_{k} |$ and $D_{j} = - \log (1 - | c_{j} |^{2})$ . Write $M_{ν} = m^{- 1} \sum_{k} c_{k}^{ν}$ for the normalised complex power sums used in this calculation. The absolutely convergent series for $\log (1 - z)$ gives

h_{j} = - ℜ \sum_{ν \geq 1} \frac{{\overset{―}{c}}_{j}^{ν} M_{ν}}{ν}, - \frac{1}{m} \sum_{j} h_{j} = \sum_{ν \geq 1} \frac{| M_{ν} |^{2}}{ν} .

In particular, $\sum_{j} h_{j} \leq 0$ . Cauchy–Schwarz applied to the first series, using $\sum_{ν \geq 1} | c_{j} |^{2 ν} / ν = D_{j}$ , gives $h_{j}^{2} \leq (- m^{- 1} \sum_{i} h_{i}) D_{j}$ . The radius bound gives $D_{j} \leq 2$ . Put $r = \sqrt{- m^{- 1} \sum_{i} h_{i}}$ . If $r = 0$ , every $h_{j}$ vanishes. For $r > 0$ , convexity bounds $e^{h}$ by the chord of its graph over $[- \sqrt{2} r, \sqrt{2} r]$ . Averaging and using $m^{- 1} \sum_{j} h_{j} = - r^{2}$ gives

\frac{1}{m} \sum_{j} e^{h_{j}} \leq \cosh (\sqrt{2} r) - \frac{r}{\sqrt{2}} \sinh (\sqrt{2} r) \leq 1.

For the last inequality, the function $\cosh u - (u / 2) \sinh u$ equals $1$ at $u = 0$ and has derivative $(\sinh u - u \cosh u) / 2 \leq 0$ for $u \geq 0$ . Thus the radius bound supplies the estimates needed for the geometric-mean inequality; no further series hypotheses are imposed. This alternative proof includes the all-zero configuration but requires the smaller radius $\sqrt{1 - e^{- 2}}$ . The weighted theorem above covers the whole closed unit disc. The next logarithmic argument imposes a different sufficient radius condition; neither restriction belongs to the weighted theorem.

For $m \geq 1$ , let $| c_{i} |^{2} \leq a < 1$ , with $a \geq 0$ , and put $L = - \log (1 - a)$ . The uniform-radius theorem gives the geometric-mean inequality

e^{L} \leq 1 + 2 L ⟹ \sum_{i = 1}^{m} {(\prod_{j = 1}^{m} | 1 - {\overset{―}{c}}_{i} c_{j} |)}^{1 / m} \leq m .

For $a = 1 / 2$ the scalar hypothesis holds, since $2 \leq 1 + 2 \log 2$ ; for $a = 3 / 4$ it fails, since $4 > 1 + 2 \log 4$ . Thus it supplies a sufficient restriction on the squared radii, not a necessary condition for the geometric-mean inequality.

The proof uses point-dependent logarithmic bounds. Write $H_{i} = m^{- 1} \sum_{j} \log | 1 - {\overset{―}{c}}_{i} c_{j} |$ and $D_{i} = - \log (1 - | c_{i} |^{2})$ . The preceding logarithmic calculation gives $\sum_{i} H_{i} \leq 0$ and $H_{i}^{2} \leq - (D_{i} / m) \sum_{j} H_{j}$ . For nonnegative upper bounds $M_{i} \geq H_{i}$ , Taylor’s formula with integral remainder gives

e^{H_{i}} \leq 1 + H_{i} + Φ (M_{i}) H_{i}^{2}, Φ (t) = \int_{0}^{1} (1 - s) e^{s t} d s .

Since $Φ (M_{i}) \geq 0$ , summing and using the quadratic bounds yields

\sum_{i} e^{H_{i}} \leq m + (1 - \frac{1}{m} \sum_{i} Φ (M_{i}) D_{i}) \sum_{j} H_{j} \leq m

whenever $\sum_{i} Φ (M_{i}) D_{i} \leq m$ . This is the sufficient condition in the formal inequality with individual upper bounds. Taking $M_{i} = L$ gives $Φ (L) L \leq 1$ as a sufficient condition, since $H_{i}, D_{i} \leq L$ . For $L > 0$ , integration gives $Φ (L) = (e^{L} - 1 - L) / L^{2}$ ; hence this condition is exactly $e^{L} \leq 1 + 2 L$ . At $L = 0$ all centres vanish, so the conclusion is immediate.

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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