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 Newton value equation

Along a Newton trajectory the polynomial value decays exponentially. We first derive this scalar equation, then use it to exclude a finite connection between critical points whose values lie on different rays. For the classical trajectory and graph statements we follow Kozen and Stefánsson [7].

Sutherland uses the term Newton flow for $\dot{z} = - f (z) / f^{'} (z)$ and notes that its solution curves map under $f$ to radial lines [6]. The value equation [7] determines their orientation and time parametrisation. The global Newtonian graph [7], described with its endpoint conventions below, is a separate input to the inverse-sheet discussion in Section 13.

A Newton trajectory is a differentiable curve satisfying

z^{'} (t) = - \frac{f (z (t))}{f^{'} (z (t))}

where $f^{'} (z (t)) \neq 0$ . Let $I \subseteq R$ be an interval on which these assumptions hold; put $w (t) = f (z (t))$ . Kozen and Stefánsson record the following identity as a lemma of Shub, Tischler and Williams [7].

Theorem 11.1 (value equation). For a polynomial $f$ and a differentiable curve $z : I \to C$ on an interval $I$ , assume $f^{'} (z (t)) \neq 0$ and $z^{'} (t) = - f (z (t)) / f^{'} (z (t))$ throughout $I$ . Then $w = f \circ z$ satisfies $w^{'} = - w$ , and

f (z (t)) = e^{- (t - t_{0})} f (z (t_{0})) (t, t_{0} \in I) .

The computation is one line: $w^{'} = f^{'} (z) z^{'} = f^{'} (z) \cdot (- f (z) / f^{'} (z)) = - f (z) = - w$ . Equivalently,

\frac{d}{d t} (e^{t} f (z (t))) = 0,

and integration on the real interval gives

f (z (t)) = e^{- (t - t_{0})} f (z (t_{0})), t_{0} \leq t, t_{0}, t \in I .

Lean proves the theorem together with this integrated identity. The separate local complex-parameter calculations are the value derivative and the scaled derivative. They require complex differentiability and do not themselves integrate a trajectory on a real interval. For an existing trajectory with nonzero initial value, its value moves inward on one positive ray; a zero value has no argument and remains zero. Thus $| f | < 1$ is preserved for later times in that trajectory’s existing interval. No global existence or description of a whole ray preimage follows from this scalar equation.

Corollary 11.2 (ray separation). Let $a < b$ and let the value trajectory $t \mapsto f (z (t))$ be continuous on $[a, b]$ . Assume the Newton equation and $f^{'} (z (t)) \neq 0$ on $(a, b)$ only. Then

f (z (b)) = e^{a - b} f (z (a)) .

If these endpoint values are nonzero, they lie on one positive ray. Therefore critical points with values on distinct positive rays cannot be endpoints of such a finite connection. The trajectory in the $z$ plane need not be radial.

The interior identity passes to the endpoints by continuity, not by evaluating $- f / f^{'}$ at a critical endpoint. The corollary constructs no trajectories and supplies no global monodromy theorem. The Lean ray exclusion assumes the exponential endpoint identity; it does not construct or integrate a connecting trajectory. Kozen and Stefánsson draw the same ray consequence for the Newtonian graph: under $f$ , every edge maps onto a segment of a ray through the origin whose endpoints are $0$ or critical values [7].

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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