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

Separating the rays of critical values

A constant perturbation translates the critical values and leaves the critical points fixed. We can therefore separate their positive rays by describing the exceptional translation parameters: two nonzero values collide in argument exactly when their ratio is positive real.

Along a Newton trajectory the argument of $f$ stays constant while its modulus decreases. By Corollary 11.2, arranging pairwise distinct arguments of the nonzero critical values is therefore a sufficient way to exclude finite connections between critical points. It is not asserted to be necessary. This condition imposes no lower bound on the distances between critical values and no upper bound on their moduli.

The weaker conditions really can coexist with a connection, even when all roots lie in the open unit disc. Take

f (z) = z^{3} - \frac{3}{16} z + \frac{3}{64} .

On $| z | = 1$ , the lower terms have total modulus at most $15 / 64 < 1$ , so Rouché’s theorem places all three roots inside. The critical points are $- 1 / 4$ and $1 / 4$ , with distinct positive values $5 / 64$ and $1 / 64$ . On the intervening real interval $f$ is positive and strictly decreasing. The equation

f (x (t)) = \frac{5}{64} e^{- t}, 0 < t < \log 5,

therefore defines a differentiable path with $x^{'} (t) = - f (x (t)) / f^{'} (x (t))$ . Its endpoint limits are $- 1 / 4$ and $1 / 4$ . It is an actual critical-to-critical Newton trajectory, not merely a pair of values on one ray. The time convention is that of the value equation [7]; the singular field is not evaluated at the endpoints.

A common translation of distinct values can separate their arguments. The next calculation describes the translations to avoid.

Theorem 12.1 (ray-collision locus). Let $a \neq b$ be complex. Every common translation $β$ for which $a + β$ and $b + β$ lie on the same positive ray has the form

β = \frac{r a - b}{1 - r}, r \in R_{> 0}, r \neq 1 .

Indeed, write $b + β = r (a + β)$ with $r > 0$ . Since $a \neq b$ , one has $r \neq 1$ , and solving for $β$ gives the displayed formula. Equivalently, $β = - a + (a - b) / (1 - r)$ , so the forbidden translations lie on the real affine line through $- a$ and $- b$ .

A finite union of these loci has empty interior. Adding a constant to $f$ translates all critical values by that constant and leaves the critical points unchanged. Section 15 records the formal avoidance and root-retention estimates. A constant cannot separate equal critical values; coefficient perturbation and control of the remaining geometric margins are separate requirements.

Local polynomial expansions

For $f (z) = \prod_{j} (z - a_{j})$ with $a_{j} \neq 0$ , put $r = min_{j} | a_{j} |$ and $p_{m} = \sum_{j} a_{j}^{- m}$ . On $| z | < r$ , the exact expansion

\log | f (z) | = \log | f (0) | - ℜ \sum_{m \geq 1} \frac{p_{m}}{m} z^{m}

expresses the local level geometry through the reciprocal power sums. If $q = | z | / r < 1$ , the tail after $N$ has modulus at most $n q^{N + 1} / ((N + 1) (1 - q))$ ; differentiated geometric series give the corresponding first and second derivative bounds. A research note gives recurrences for these reciprocal power sums and tests based on the first two derivatives for a locally contained piecewise path. The assumption $| z | < r$ is essential: this disc contains no zero of $f$ , so a logarithm exists there. The expansion does not continue through a root and does not select a pair of roots to join. A proposed local path must still satisfy the containment inequalities.

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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