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 mean bound for critical values in every degree

The Poisson argument first gives a quadratic mean bound, allowing a disc of any centre and radius and counting all multiplicities. This is the formally supported theorem below. The subsequent fourth-power refinement, and the higher exponents obtained from vanishing complex power sums, are unformalised analytic remarks. A critical-value moment bound supplies no estimate for the lengths of inverse paths ending at those points.

Dubinin [21] proves the sharp unit-disc critical-value product inequality using the resultant identity, the maximum-modulus principle and Schur’s Vandermonde inequality. The positive-moment estimate below implies his product bound by AM–GM. When $n \geq 3$ , the product estimate alone cannot control a positive moment: a list $T, T^{- 1}, 1, \dots, 1$ has product one, but its sum of $p$ th powers is unbounded as $T \to \infty$ for every $p > 0$ . This compares the information in two inequalities for nonnegative lists; it does not claim that these lists occur as critical values of polynomials in the theorem. In degree two there is just one critical value, and the product and positive-moment bounds are equivalent. For a zero prescribed at the origin, his Theorem 3 (p. 1174) gives

{(\prod_{j} | f (c_{j}) |)}^{1 / (n - 1)} \leq (n - 1) {(\frac{| f^{'} (0) |^{2}}{n^{n}})}^{1 / (n - 1)} .

Dubinin explicitly identifies Tischler’s earlier contribution [25]. His Theorem 1 uses dissymmetrisation; that is not the proof mechanism of Theorem 2. Schur’s original paper [27] is the historical antecedent cited there, not a separately re-proved source in this record. A sharp marked-zero positive-moment bound remains a separate question. The addendum [26] reports a reduction to boundary-root configurations, without solving the resulting angular optimisation. That unavailable addendum is not a proof dependency of the critical-value theorem below.

Theorem 10.2 (a mean bound for critical values). Let $f$ be monic of degree $n \geq 2$ , with roots in a closed disc of radius $R \geq 0$ . If $c_{1}, \dots, c_{n - 1}$ are its critical points counted with multiplicity, then

\begin{matrix} (11) & \sum_{j = 1}^{n - 1} | f (c_{j}) |^{2 / (n - 1)} \leq (n - 1) R^{2 n / (n - 1)} . \end{matrix}

Consequently the lower exponents used elsewhere in the record satisfy

\begin{matrix} (12) & \sum_{j = 1}^{n - 1} | f (c_{j}) |^{1 / (n - 1)} \leq (n - 1) R^{n / (n - 1)} \end{matrix}

and

\sum_{j = 1}^{n - 1} | f (c_{j}) |^{1 / n} \leq (n - 1) R .

The constant is attained by $f (z) = (z - τ)^{n} - λ$ with enclosing disk centred at $τ$ and radius $R = | λ |^{1 / n}$ .

The finite inequality behind the theorem concerns arbitrary points of the disk, without asking them to arise as critical points. Its quadratic form is

\sum_{j = 1}^{m} {(\prod_{k = 1}^{m} | 1 - \overset{―}{c_{j}} c_{k} |)}^{2 / m} \leq m, | c_{j} | \leq 1.

The following pointwise bound explains why these finite inequalities control critical values in every degree.

Lemma 10.3 (reflected-derivative bound). If $f$ is monic of degree $n \geq 2$ with roots in the closed unit disk, and $c_{1}, \dots, c_{n - 1}$ list its critical points with multiplicity, then

\begin{matrix} (13) & | f (c_{j}) | \leq \prod_{k = 1}^{n - 1} | 1 - \overset{―}{c_{j}} c_{k} | . \end{matrix}

Proof. First put all roots $a_{i}$ strictly inside the disk. We want a numerator that equals $n f$ at the critical points and can be compared with $f^{'}$ on the circle. Set

N (z) = n f (z) - z f^{'} (z), G (z) = n \prod_{k} (1 - \overset{―}{c_{k}} z) .

The numerator is the polar derivative at $0$ : the conventional polar derivative at $α$ is $n f (z) + (α - z) f^{'} (z)$ ; see Rather–Gulzar, Section 1 for this notation. Its leading term cancels, and $N (c_{j}) = n f (c_{j})$ . Reflection gives $| G | = | f^{'} |$ on the circle. Gauss–Lucas puts every $c_{k}$ inside the disc, so $G$ has no zero on its closure. Thus $N / G$ is a holomorphic quotient to which the maximum-modulus principle applies. On $| ζ | = 1$ ,

Re \frac{ζ f^{'} (ζ)}{f (ζ)} = \sum_{i} (\frac{1}{2} + \frac{1 - | a_{i} |^{2}}{2 | 1 - a_{i} \bar{ζ} |^{2}}) \geq \frac{n}{2} .

Writing the logarithmic derivative as $w$ , the identity $| n - w |^{2} - | w |^{2} = n^{2} - 2 n Re w \leq 0$ gives $| N | \leq | G |$ on the circle. The maximum-modulus principle applied to $N / G$ gives the same comparison inside. At $c_{j}$ , $N (c_{j}) = n f (c_{j})$ ; dividing by $n$ and conjugating each factor proves (13). For closed-disk roots apply this result to $f_{r} (z) = r^{n} f (z / r)$ , $0 < r < 1$ , whose critical points are $r c_{k}$ :

r^{n} | f (c_{j}) | \leq \prod_{k} | 1 - r^{2} \overset{―}{c_{j}} c_{k} | .

Let $r ↑ 1$ . This also handles boundary critical points and all multiplicities without selecting local branches. ◻

First suppose the zeros of $f$ lie in the closed unit disc, and set $m = n - 1$ . Gauss–Lucas puts the critical points in the same disc. Lemma 10.3 gives the first inequality below; the weighted Poisson inequality, proved next and applied with equal weights $1 / m$ , gives the second:

\sum_{j = 1}^{m} | f (c_{j}) |^{2 / m} \leq \sum_{j = 1}^{m} {(\prod_{k = 1}^{m} | 1 - \overset{―}{c_{j}} c_{k} |)}^{2 / m} \leq m .

Applying this to $R^{- n} f (h + R z)$ and scaling back proves (11) for $R > 0$ ; if $R = 0$ , all critical values vanish.

The two displayed consequences are the lower power means of the same nonnegative $m$ -tuple. Equivalently, after unit-disc normalization, apply the monotonicity of normalized $L^{p}$ means from exponent $2 / m$ first to $1 / m$ and then to $1 / (m + 1) = 1 / n$ . Scaling back contributes respectively $R^{n / m}$ and $R$ . These are consequences of the quadratic case; the higher-power argument below strengthens that case without changing these lower-exponent applications.

The quadratic moment controls the number of large critical values. For $R > 0$ , write $r_{j} = | f (c_{j}) |^{1 / n}$ . Then

\sum_{j} (r_{j} / R)^{2 n / (n - 1)} \leq n - 1, # {j : r_{j} \geq t R} \leq \frac{n - 1}{t^{2 n / (n - 1)}} (t > 0) .

The counting bound follows by retaining just the indicated summands. The fourth-power refinement proved below replaces the exponent $2 n / (n - 1)$ here by $4 n / (n - 1)$ , improving the count for $t > 1$ . For $0 < t \leq 1$ the trivial bound $n - 1$ is at least as good. The binomial equality family has every $r_{j} = R$ , but these distributional bounds still leave the geometry of the joining paths to be supplied.

The weighted Poisson inequality and its equality case.

Higher powers and vanishing complex power sums.

Lean proofs of the Poisson and integration steps.

A uniform central region in every degree.

Four points without a matching or numerical split.

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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