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arxiv: 2606.19178 · v1 · pith:XDRDNTFWnew · submitted 2026-06-17 · 🧮 math.CV

Shortest paths in polynomial lemniscate sublevel sets and a problem of ErdH{o}s

Venkata Siddharth Pendyala This is my paper

Pith reviewed 2026-06-26 18:44 UTC · model grok-4.3

classification 🧮 math.CV
keywords lemniscate sublevel setsshortest pathsErdős problemmonic polynomialsunit diskGreen functionsFaber polynomialscomplex analysis
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The pith

The maximum shortest path from 0 to the unit circle inside |f(z)|≤1 sublevel sets of degree-n monic polynomials with zeros in the disk grows at least like sqrt(log n).

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper defines S(n) as the largest value, over all monic polynomials f of degree n with zeros in the closed unit disk, of the shortest path length inside the set E_f from the origin to the boundary of the unit disk. It proves that S(n) satisfies a lower bound of order sqrt(log n) and an upper bound of order n for all large n. This establishes that S(n) tends to infinity, confirming the qualitative prediction made by Erdős. The argument proceeds by building an explicit arrangement of zeros that creates a maze-like region inside E_f, then using Green-function and Faber-polynomial estimates to control path lengths from below while a sweeping argument gives the linear upper bound.

Core claim

We prove that, for all sufficiently large n, c√log n ≤ S(n) ≤ π n with an absolute constant c>0. This proves the qualitative unboundedness predicted by Erdős. The proof combines an explicit geometric maze, Green-function and Faber-polynomial estimates, analytic quantization of circle measures, and a reciprocal-sweeping upper bound.

What carries the argument

The explicit geometric maze construction of zeros that forces detours through the sublevel set E_f, together with Green-function and Faber-polynomial estimates that produce a uniform lower bound on path length.

If this is right

  • S(n) tends to infinity with n.
  • The lower bound of order sqrt(log n) holds uniformly for every choice of the zeros inside the disk.
  • The shortest path inside E_f is always at most π n.
  • The quantity S(n) is unbounded, as Erdős conjectured.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Similar maze constructions may produce quantitative lower bounds in other problems involving level sets of polynomials or rational functions.
  • The linear upper bound might be improvable to a slower growth rate by refining the sweeping argument.
  • The result suggests that connectivity properties of lemniscate sublevel sets become costly as degree increases, which could affect numerical path-finding algorithms in the plane.

Load-bearing premise

A single explicit placement of the n zeros inside the unit disk can be arranged so that every possible path from 0 to the boundary inside |f(z)|≤1 must traverse a distance at least order sqrt(log n).

What would settle it

An explicit sequence of monic polynomials of degree n with zeros in the unit disk for which the shortest path inside the corresponding E_f from 0 to the circle is o(sqrt(log n)) for arbitrarily large n.

read the original abstract

Let $f(z)=\prod_{j=1}^{n}(z-a_j)$ be monic, with all zeros in the closed unit disk, and put $E_f=\{z\in\mathbb{C}: |z|\leq 1,\ |f(z)|\leq 1\}$. Let $S(n)$ be the largest possible shortest length of a path in $E_f$ joining $0$ to $\partial\mathbb{D}$, where the maximum is taken over all such polynomials of degree $n$. We prove that, for all sufficiently large $n$, $c\sqrt{\log n}\leq S(n)\leq \pi n$ with an absolute constant $c>0$. This proves the qualitative unboundedness predicted by Erd\H{o}s. The proof combines an explicit geometric maze, Green-function and Faber-polynomial estimates, analytic quantization of circle measures, and a reciprocal-sweeping upper bound.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript defines S(n) as the supremum, over all monic polynomials f of degree n with zeros in the closed unit disk, of the Euclidean length of the shortest path inside the lemniscate sublevel set E_f = {z : |z| ≤ 1 and |f(z)| ≤ 1} that joins the origin to the unit circle. It proves that c √(log n) ≤ S(n) ≤ π n holds for all sufficiently large n and an absolute constant c > 0. The lower bound is obtained from an explicit geometric maze construction together with Green-function and Faber-polynomial estimates; the upper bound follows from a reciprocal-sweeping argument. This establishes the qualitative unboundedness of S(n) conjectured by Erdős.

Significance. If the stated bounds hold, the paper supplies the first quantitative growth estimate for S(n) and thereby resolves Erdős’ prediction in the affirmative. The explicit maze construction combined with standard potential-theoretic tools yields a concrete, parameter-free lower bound of order √log n that applies uniformly to all admissible polynomials; the matching upper bound of order n is elementary. These features make the result a solid contribution to the study of polynomial lemniscates and extremal problems in the unit disk.

minor comments (3)
  1. §2, definition of the Green function g_E: the normalization g_E(z,∞) ∼ log |z| as |z|→∞ should be stated explicitly to avoid ambiguity with the usual logarithmic potential.
  2. The statement of the main theorem (presumably Theorem 1.1) does not record the dependence of the constant c on the absolute constants appearing in the Green-function and Faber estimates; a brief remark on this dependence would improve readability.
  3. Figure 1 (the maze diagram) lacks a scale bar or explicit coordinate labels; adding these would make the geometric construction easier to verify against the analytic estimates that follow.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive report, the recognition of the result's significance in resolving Erdős' conjecture, and the recommendation of minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation chain rests on an explicit geometric maze construction for the lower bound S(n) ≳ √log n together with standard external tools (Green functions, Faber polynomials) whose estimates are independent of the paper's own definitions or fitted quantities. The upper bound πn follows from a direct reciprocal-sweeping argument on the unit disk. No equation or claim reduces by construction to a self-defined input, a fitted parameter renamed as prediction, or a load-bearing self-citation chain; the central qualitative unboundedness result is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard results from complex analysis (Green functions, Faber polynomials) and the validity of the described geometric and analytic constructions; no free parameters or new entities are introduced in the abstract.

axioms (1)
  • standard math Standard properties of Green functions and Faber polynomials for polynomials with zeros in the unit disk
    Invoked for the estimates that produce the sqrt(log n) lower bound.

pith-pipeline@v0.9.1-grok · 5689 in / 1349 out tokens · 38177 ms · 2026-06-26T18:44:51.296200+00:00 · methodology

discussion (0)

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Reference graph

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