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

Power substitutions in trinomials

The segment identity survives a power substitution. We record the lifting argument as an unformalised remark: the source checks the segment inequality and finite length calculation, but not selection and lifting of a suitable quotient root. Fix integers $1 \leq r < m$ and $q \geq 1$ , and consider the translated monic polynomial

f (z) = (z - h)^{q m} + a (z - h)^{q r} + c .

The strict exponent inequality keeps the leading monomial separate. The algebraic identity below also holds when $r = m$ , but then the two leading terms merge and the displayed polynomial need not be monic or have degree $q m$ . Writing $w = (z - h)^{q}$ reduces the root equation to $w^{m} + a w^{r} + c = 0$ . At such a quotient root the middle coefficient can be eliminated exactly: for $0 \leq u \leq 1$ ,

u^{m} w^{m} + a u^{r} w^{r} + c = (1 - u^{r}) c - (u^{r} - u^{m}) w^{m} .

This is a nonnegative combination of $c$ and $- w^{m}$ , with weights $1 - u^{r}$ and $u^{r} - u^{m}$ summing to $1 - u^{m}$ . The formal radial-segment estimate gives strict containment when $| w | < 1$ and $| c | < 1$ .

For $q \geq 2$ and a nonzero quotient root satisfying these bounds, choose two distinct solutions $y_{1}, y_{2}$ of $y^{q} = w$ . Their common modulus is $| w |^{1 / q} < 1$ . To lift a quotient radial segment, set $u = t^{q}$ :

| f (h + t y_{i}) | \leq (1 - t^{q m}) max {| c |, | w |^{m}} < 1 (0 \leq t \leq 1) .

Thus the two lifted segments join $h + y_{1}$ to $h + y_{2}$ through $h$ with length $2 | w |^{1 / q} < 2$ . The construction uses any quotient root with the stated modulus bound; it does not prove that such a root exists without an additional hypothesis. A zero quotient root instead gives a root of multiplicity at least $q$ at $h$ and hence a constant path between two occurrences. The formal source checks the factorization, radial-segment estimate and finite length inequality. It does not establish the existence of a suitable quotient root, formalise the selection and lifting just described, or prove unrestricted Erdős #1041.

The radial-segment identity and strict bound are formalised in cancellation along a trinomial root segment and containment of a trinomial root segment.

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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