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 exact binomial chord calculation

For $n \geq 2$ and $0 < | a | < 1$ , put $r = | a |^{1 / n}$ and $r_{*} = (1 + \cos^{n} (π / n))^{- 1 / n}$ . A rotation reduces $z^{n} - a$ to $z^{n} - r^{n}$ without changing lengths or moduli. For $n = 2$ , $r_{*} = 1$ and only the outer-diameter case occurs.

Theorem 10.1 (complementary binomial chords). Two adjacent zeros of $z^{n} - a$ can be joined by an explicit polygonal path inside ${| z^{n} - a | < 1}$ of length strictly below $2$ . For $r < r_{*}$ the adjacent-root chord itself works. For $r \geq r_{*}$ , two radial legs and an inner adjacent crossing chord work after an arbitrarily small radial contraction. These two constructions meet at $r = r_{*}$ , where the outer chord attains $| f | = 1$ at its midpoint and therefore lies only in the closed lemniscate. Open containment at and above the switch uses the inner chord after a radial contraction.

For $f (z) = z^{n} - r^{n}$ , $0 < r < 1$ and $n \geq 2$ , the two constructions depend on the maximum over an entire chord. The endpoint values alone do not give that maximum. Put $ω = e^{2 π i / n}$ and $c = \cos (π / n)$ . When $n \geq 3$ , define $r_{*} = (1 + c^{n})^{- 1 / n}$ and $ε = (1 - r^{n})^{1 / n}$ ; in the inner-chord case $r \geq r_{*}$ , set $t = ε / c \leq r$ .

We prove the chord estimate also at the smaller radii needed for the second construction. For $n \geq 3$ , let $0 < s \leq r$ . A point on the chord from $s$ to $s ω$ can be written

z = \frac{s c}{\cos θ} e^{i (π / n + θ)}, | θ | \leq π / n .

Concavity of $\log \cos x$ on $[0, π / 2)$ gives $\cos (n θ / 2) \leq \cos^{n / 2} θ$ for $0 \leq θ < π / n$ . By symmetry and continuity,

1 + \cos (n θ) \leq 2 \cos^{n} θ (| θ | \leq π / n) .

Set $u = \cos^{- n} θ \geq 1$ . Since $z^{n} = - (s c)^{n} u e^{i n θ}$ , this inequality gives

\begin{aligned} | z^{n} - r^{n} |^{2} - (r^{n} + (s c)^{n})^{2} & \leq (u - 1) (s c)^{n} ((s c)^{n} (u + 1) - 2 r^{n}) \leq 0. \end{aligned}

The last step uses $(s c)^{n} u \leq s^{n} \leq r^{n}$ and $(s c)^{n} \leq r^{n}$ . Equality holds at the midpoint, where $θ = 0$ . Consequently the maximum on every such chord is exactly $r^{n} + (s c)^{n}$ . Taking $s = r$ proves the outer threshold; taking $s = t = ε / c \leq r$ proves the inner bound, and taking $s = λ t < t$ makes it strict. Every radial leg between $s$ and $r$ has modulus $r^{n} - | z |^{n} < 1$ . Its two legs and crossing chord have length $2 r - 2 s (1 - \sin (π / n)) < 2 r$ , which proves the stated length formula and shows that contraction preserves the strict length bound.

For $n = 2$ , the chord is a diameter and its maximum is $r^{2} < 1$ ; its length is $2 r < 2$ . For $n \geq 3$ , the outer chord works exactly when $r < r_{*}$ , and its length is $2 r \sin (π / n)$ . At $r = r_{*}$ it reaches level one at its midpoint. If $r \geq r_{*}$ , the two radial legs and inner crossing have length

2 r - 2 ε \tan (\frac{π}{4} - \frac{π}{2 n}) < 2 r < 2.

Contracting the crossing radius makes the level bound strict, as proved above. The comparison concerns these two constructions only; it makes no claim about shortest paths among all admissible curves.

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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