Erdős #114 — max lemniscate length is z^n−1

Statement

If $p(z)\in\mathbb{C}[z]$ is a monic polynomial of degree $n$, is the length of the curve (lemniscate) $\{z\in\mathbb{C}: |p(z)|=1\}$ maximised when $p(z)=z^n-1$? I.e., writing $f(n)$ for the maximal such length over all monic degree-$n$ $p$, is $f(n) = \ell(\partial E_1(z^n-1))$ for every $n$, where $\ell(\partial E_1(z^n-1)) = 2^{1/n}B(\tfrac12,\tfrac1{2n}) = 2n+4\log 2+O(1/n)$ ($B$ = Euler beta function)?

Facts

- Prize \$250; site status Open, FALSIFIABLE (erdosproblems.com/114, page last edited 23 Jan 2026 — reflects the site owner's belief that the *exact, all-$n$* form of the conjecture is still open, even though the asymptotic form is now proved; see Literature state). - Falsifiable: yes in principle — a single $n$ and a monic degree-$n$ $p\neq z^n-1$ (up to rotation/translation) with strictly longer lemniscate would refute it for that $n$. - Origin: Erdős, Herzog, Piranian, "Metric properties of polynomials," J. Analyse Math. 7 (1958) [EHP58, Problem 12]; restated [Er61,p.247] [Ha74, Problem 4.10] [Er82e] [Er90] [Er97f] [Va99,2.35]. - Known results / best bounds (chronological, all verified against erdosproblems.com/114 remarks and cross-checked in Tao's Table 1, arXiv:2512.12455): - Pommerenke [Po59] proved the companion *lower*-bound question from [EHP58]: if $\{z:|p(z)|<1\}$ is connected then $\ell \geq 2\pi$, tight for $p=z^n$. - Pommerenke [Po61]: $f(n) \ll n^2$. - Dolzhenko [Do61]: $f(n)\leq 4\pi n$ (little-known at the time). - Borwein, "The arc length of the lemniscate $\{|p(z)|=1\}$," Proc. AMS 123 (1995) [Bo95]: $f(n) \ll n$ (unaware of Dolzhenko); reports the \$250 prize. - Eremenko–Hayman, "On the length of lemniscates," Michigan Math. J. 46 (1999) [ErHa99]: proves the *full* conjecture for $n=2$; $f(n)\leq 9.173n$ in general; also proves existence of an extremal polynomial whose lemniscate is connected and contains all critical points (used by every later paper). - Danchenko [Da07]: $f(n)\leq 2\pi n$. - Kosukhin, Math. Notes 92 (2012): $f(n) \leq \pi n + O(\sqrt{n\log n})$. - Fryntov–Nazarov, "New estimates for the length of the Erdos-Herzog-Piranian lemniscate," in *Linear and complex analysis* (AMS, 2009) [FrNa09]: proves $z^n-1$ is a local maximiser, and the asymptotic bound $f(n) \leq 2n + O(n^{7/8})$ — writing "we have no doubt the power $7/8$ can be substantially improved though to bring it below $1/2$ seems quite a challenging problem." - Terence Tao, "The maximal length of the Erdős–Herzog–Piranian lemniscate in high degree," arXiv:2512.12455 (Dec 2025): building on Fryntov–Nazarov, proves $f(n)\leq 2n+O(\sqrt n)$, then $2n+O(1)$, then $2n+4\log2+o(1)$, and finally the full conjecture for all sufficiently large $n$, with $z^n-1$ the *unique* maximiser up to rotation/translation (Theorem 1.1(i)-(iv)). Constants are effectively computable. - Formalized in Lean: No (site field "Formalised statement?" = No as of fetch). - Related problems: erdos/115 erdos/116 — both from the same Erdős–Herzog–Piranian 1958 paper on metric properties of $\{|p(z)| \lessgtr 1\}$, both now solved (see Related).

Literature state

Not fully resolved (the exact all-$n$ statement is still officially open), but overwhelmingly resolved asymptotically, and only a finite computation stands between the current state of the art and full resolution.

- Tao's Dec-2025 paper (arXiv:2512.12455, math.CV, 56pp) is the decisive advance: Theorem 1.1(iv) proves $p(z)=z^n-1$ (up to symmetry) is the unique length-maximiser for all $n \geq n_0$ for some unspecified but *effectively computable* $n_0$. Method: normalize $p$ (translate/rotate so $z^{n-1}$-coeff vanishes, $p(0)\leq0$ real), use Eremenko–Hayman's existence-of-a-connected-critical-point-containing extremizer, then measure deviation from $p_0=z^n-1$ via the $\ell^1$ "dispersion" of critical points $\|p\|_1=\sum_\zeta|\zeta|$ and an "origin repulsion" $\|p\|_0 = n|1+p(0)|^{1/n}$; the heuristic (rigorously established via a long stability/perturbation analysis, quasiconformal-mapping tools, and Riemann–Hurwitz-formula counting used in the Eremenko–Hayman existence lemma) is $\ell(\partial E_1(p)) \approx \ell(\partial E_1(p_0)) - c\|p\|$ for a constant $c>0$. Tao notes explicit resemblance to his own earlier proof of Sendov's conjecture for sufficiently high degree (Acta Math. 229 (2022), 347–392) — same "sufficiently large degree + effective local-stability perturbation" strategy, though the technical machinery differs. - Remark 1.3 of the Tao paper is the key attack-surface fact: "the full verification of Conjecture 1.1 now reduces to checking the conjecture for an explicitly bounded number of $n$" — i.e. this problem is now *almost decidable*, blocked only by (a) Tao not computing an explicit $n_0$, and (b) a theoretical (very unlikely) obstruction if a rival extremizer's lemniscate length were exactly (transcendentally) tied with $\ell(\partial E_1(z^n-1))$ at some small $n$. - Tao's numerical exploration for the paper used AlphaEvolve (DeepMind's LLM-driven evolutionary code-search tool) plus Gemini to generate plotting code and to numerically confirm the conjecture held for many small-to-moderate $n$ before the proof was found — an explicit case of AI-assisted numerical exploration feeding a human/Terence-Tao-authored rigorous proof (stated in the paper's Fig. 1 caption and §1.1). - Unverified community claims (forum, not incorporated into the site's official remarks, not peer-reviewed): On erdosproblems.com/forum/discuss/114, user "dahlkebj" posted a manuscript (Zenodo DOI 10.5281/zenodo.20318611) claiming a sharp proof for the cubic case $n=3$ via the Eremenko–Hayman reduction + Chebyshev-$T_3$/Joukowski-map identity. User "Kenneth Mendoza" (KMendoza) posted (Zenodo DOI 10.5281/zenodo.19229245, later superseded by 10.5281/zenodo.20087919) a claimed certified interval-arithmetic (IEEE 1788) branch-and-bound computational verification of the conjecture for $3\leq n\leq 12$, later extended by "dahlkebj" to $1\leq n\leq14$ (dual Python/mpmath + Rust/inari implementations, code at github.com/MendozaLab/erdos-experiments). These are self-published, not journal- or arXiv-reviewed, and explicitly not yet counted by the site ("There are no solutions, partial or complete, claimed in the comments" banner) — treat as an unverified lead, not an established fact, until independently checked or published. If genuine, combined with Tao's asymptotic theorem this would leave only a finite explicit gap $15\leq n<n_0$ to close. - No indication any AI system (other than AlphaEvolve as a numerics-exploration aid to Tao) has produced new *proved* mathematics on this problem; no Lean/formal-conjectures entry found for #114 specifically (unlike #115, which is Lean-formalized/proved).

Attack surface

- Mode: finite-search (the realistic near-term win) + literature-resolution/derivation (extracting/tightening Tao's effective $n_0$, and independently verifying or reproducing the forum's interval-arithmetic small-$n$ claims). - Concrete first experiment: (1) Read Tao's proof (arXiv:2512.12455) closely enough to extract or re-derive an explicit, if crude, value of $n_0$ from the effective constants in Theorem 1.1(i)-(iv) (the paper states all constants are "effectively computable" but does not optimize them). (2) Independently reproduce a certified branch-and-bound / interval-arithmetic (e.g. Rust inari, or Arb/arblib, or Julia IntervalArithmetic.jl) search over normalized monic degree-$n$ polynomials (parametrized by the $n-1$ critical points via $p'(z)=n\prod(z-\zeta)$ and $p(0)$, as in Tao's own parametrization) for each $3\leq n\leq \sim 20$, verifying $\ell(\partial E_1(p)) \leq \ell(\partial E_1(z^n-1))$ with a certified global-optimization / branch-and-bound closure — directly checkable against (and a good-faith audit of) the unverified MendozaLab/dahlkebj Zenodo claims. - Oracle: fully mechanical for any fixed $n$ — $\ell(\partial E_1(p))$ is a contour-integral of $1/|p'(z)|$ over the level set $|p(z)|=1$ (Tao's Lemma 3.1 formula), which can be rigorously bounded via certified interval arithmetic / branch-and-bound over the compact parameter space of normalized degree-$n$ critical-point configurations; a certified UNSAT-style closure (all boxes eliminated) constitutes a machine-checkable proof for that $n$, exactly as the (unverified) forum claims purport to do. - Feasibility: the exact all-$n$ statement is no longer "famous-and-hard" in the way it was before Dec 2025 — Tao's theorem converts it into essentially a finite verification problem for $n$ below an (unspecified) threshold. Genuine, reachable wins: (a) pin down Tao's implicit $n_0$ from the paper's constants; (b) build/independently verify a certified interval-arithmetic closure for all $n$ up to whatever is tractable (the forum claims $n\leq14$ is already done, unverified); (c) if $n_0$ can be brought down to meet the certified range, the \$250 problem is fully closed. This is a much more tractable target now than a typical open Erdős problem — the hard analytic work is already done by Tao.

Related

- erdos/115 — same Erdős–Herzog–Piranian family (bound on $\max|p'(z)|$ over $\{|p(z)|\leq1\}$ when that set is connected); PROVED (LEAN): Eremenko–Lempert (1994) showed Chebyshev polynomials are the exact extremizers, giving $(\tfrac12+o(1))n^2$. Same "identify + prove exact extremal polynomial in a family of level-set functionals of monic polynomials" template as #114. - erdos/116 — same EHP58 paper: lower bound on $|\{z:|p(z)|<1\}|$ for $p(z)=\prod(z-z_i)$, $|z_i|\leq1$; PROVED: lower bound $\gg(\log n)^{-1}$ via Krishnapur–Lundberg–Ramachandran, arXiv:2503.18270 (2025), improving Pommerenke [Po61] and Wagner [Wa88]'s $\gg n^{-4}$/$\ll(\log\log n)^{-1/2+\epsilon}$ bounds; sibling area/inradius extremal-polynomial-level-set problem, solved via potential-theoretic/harmonic-measure estimates. - solved/sendov-conjecture-high-degree — Tao, Acta Math. 229 (2022), 347–392: Sendov's conjecture (critical points of a polynomial with all roots in the unit disk lie within distance 1 of some root) proved for sufficiently high degree. Same author, same overarching strategy (reduce to sufficiently-large-$n$ via an effective local-stability/perturbation analysis around a conjectured extremal/critical configuration), explicitly cited by Tao himself in §1.3 of arXiv:2512.12455 as methodologically resembling #114's proof (though the technical tools differ). - concept/quasiconformal-mappings — used (via Eremenko–Hayman's existence lemma for the maximizer) as one of the two non-elementary tools underlying the setup Tao's proof relies on. - concept/riemann-hurwitz-formula — the other non-elementary tool in the Eremenko–Hayman existence-of-maximizer lemma (Proposition 1.2 of arXiv:2512.12455). - concept/critical-point-dispersion — Tao's $\ell^1$ "dispersion" $\|p\|_1=\sum_\zeta|\zeta|$ and "origin repulsion" $\|p\|_0$, the quantitative deviation-from-extremizer framework driving the whole proof (Heuristic 1.1 in the paper). - concept/local-maximiser-stability — Fryntov–Nazarov's local-maximiser/second-variation argument that Tao's global result builds on. - concept/beta-function-asymptotics — exact closed form $\ell(\partial E_1(z^n-1))=2^{1/n}B(\tfrac12,\tfrac1{2n})$ and its Stirling-type asymptotic expansion $2n+4\log2+O(1/n)$. - concept/ai-assisted-numerical-exploration — Tao used DeepMind's AlphaEvolve + Gemini for numerical exploration/plotting that guided (but did not itself prove) the theorem. - concept/interval-arithmetic-certified-computation — the certified branch-and-bound (IEEE 1788, Rust inari) approach used in the (unverified) forum claims to close small-$n$ cases; the natural finite-search oracle for the remaining gap. - concept/chebyshev-extremal-polynomials — the extremal-object template shared with #115 (Eremenko–Lempert) and invoked in the (unverified) forum's cubic-case manuscript.