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

Polynomial and roots

The perturbation of $z^{7} - 1$ is small on the unit circle but has a nontrivial shape at $z = ε w$ : the terms $ε^{4} a z^{3}$ , $ε^{5} b z^{2}$ and $ε^{6} c_{0} z$ all become order $ε^{7}$ there. The paired conjugate terms give the antisymmetry used to locate the roots. These two features let us control roots near the unit circle and critical points near the origin by different tests.

All constants are rational, so both sets of tests can be made without rounding. Set

s = 10^{- 6}, ε = s^{2} = 10^{- 12}, ρ = 1 - s^{16} = 1 - 10^{- 96},

A_{0} = - \frac{329507}{1600}, B_{0} = \frac{551827}{800}, C_{0} = - \frac{23013813}{32000},

and define

a = A_{0} - i s, b = i B_{0} + \frac{9}{5} s, c_{0} = - C_{0} - \frac{162}{25} i s .

The polynomial is

\begin{aligned} (1) & F (z) & = z^{7} - 1 + ε^{4} (a z^{3} - \bar{a} z^{4}) + ε^{5} (b z^{2} - \bar{b} z^{5}) + ε^{6} (c_{0} z - {\bar{c}}_{0} z^{6}), \\ (2) & f (z) & = ρ^{7} F (z / ρ) . \end{aligned}

The identity $F (z) = - z^{7} \overset{―}{F (1 / \bar{z})}$ is useful, but does not by itself place the zeros on the unit circle. For that purpose put $χ (x) = (1 + i x) / (1 - i x)$ . Direct expansion gives $(1 - i x)^{7} F (χ (x)) = 2 i H (x)$ , where

\begin{aligned} H (x) = & 7 x - 35 x^{3} + 21 x^{5} - x^{7} - ε^{4} (1 + x^{2})^{3} (s + A_{0} x) \\ + ε^{5} (1 + x^{2})^{2} [B_{0} (1 - 3 x^{2}) + (9 s / 5) (x^{3} - 3 x)] \\ + ε^{6} (1 + x^{2}) [C_{0} (5 x - 10 x^{3} + x^{5}) - (162 s / 25) (1 - 10 x^{2} + 5 x^{4})] . \end{aligned}

The signs at the following rational endpoints are obtained by substitution. Both endpoints in each row are divided by $10^{4}$ .

$j$	Left endpoint	Right endpoint	Left sign	Right sign
0	$- 1$	1	$-$	$+$
1	4815	4816	$+$	$-$
2	12539	12540	$-$	$+$
3	43812	43813	$+$	$-$
4	$- 43813$	$- 43812$	$+$	$-$
5	$- 12540$	$- 12539$	$-$	$+$
6	$- 4816$	$- 4815$	$+$	$-$

The seven intervals give distinct real zeros $x_{j}$ of $H$ , so $ζ_{j} = χ (x_{j})$ are all seven zeros of $F$ . They have modulus one. Hence the zeros $b_{j} = ρ ζ_{j}$ of $f$ are distinct and lie in the open unit disc.

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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