A Degree-Four Lemniscate Path Theorem

Venkata Siddharth Pendyala

arxiv: 2606.24875 · v1 · pith:TNZXCOYHnew · submitted 2026-06-23 · 🧮 math.CV

A Degree-Four Lemniscate Path Theorem

Venkata Siddharth Pendyala This is my paper

Pith reviewed 2026-06-25 21:23 UTC · model grok-4.3

classification 🧮 math.CV

keywords lemniscatepolynomial zerosunit diskmonic polynomialdegree fourpolygonal pathErdős-Herzog-Piranian problemsublevel set

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The pith

If f is monic of degree four with all zeros inside the open unit disk, then two zeros join inside {|f(z)|<1} by a polygonal path of length less than 2.

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

The paper proves the degree-four case of the Erdős-Herzog-Piranian path problem. For any monic polynomial f of degree four whose zeros all lie in the open unit disk, at least two of those zeros can be joined by a possibly degenerate polygonal path that stays entirely inside the region where |f(z)| is less than one and whose length is strictly less than two. A sympathetic reader would care because the result gives a concrete geometric control on how the sublevel sets of low-degree polynomials connect their roots, which bears on the shape of lemniscates and the distribution of zeros. The argument proceeds by examining the possible placements of four roots inside the disk and constructing explicit short paths in each configuration.

Core claim

If f is monic of degree four and all zeros of f, counted with multiplicity, lie in the open unit disk, then two zeros from this list can be joined inside {z : |f(z)| < 1} by a possibly degenerate polygonal path of length less than 2.

What carries the argument

The sublevel set {z : |f(z)| < 1} together with explicit polygonal paths connecting pairs of zeros inside it, whose length is controlled by case analysis on root positions.

If this is right

The sublevel set {|f(z)| < 1} is path-connected between at least one pair of zeros with a uniform length bound independent of the specific root locations.
Degenerate paths cover the case of multiple roots at the same point.
The result supplies an explicit constant (less than 2) for the degree-four instance of the original path problem.
Any configuration of four roots inside the disk forces at least one short connection inside the lemniscate.

Where Pith is reading between the lines

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

The same length bound may hold after suitable rescaling for non-monic polynomials of degree four.
Numerical sampling of random root placements inside the disk could locate configurations that approach the length bound of 2.
The technique of exhaustive case analysis on root positions might extend to degree five if additional combinatorial tools are introduced.

Load-bearing premise

The polynomial must be monic so that the level set |f(z)| < 1 scales correctly with the leading coefficient fixed at one.

What would settle it

Exhibit a monic degree-four polynomial whose zeros all lie inside the open unit disk such that every pair of zeros requires a connecting polygonal path of length at least 2 inside {|f(z)| < 1}.

read the original abstract

We prove the degree-four case of a path problem of Erd\H{o}s, Herzog, and Piranian. If $f$ is monic of degree four and all zeros of $f$, counted with multiplicity, lie in the open unit disk, then two zeros from this list can be joined inside $$\{z:|f(z)|<1\}$$ by a possibly degenerate polygonal path of length less than $2$.

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.

Desk Editor's Note private letter to a colleague

The paper settles the degree-four case of the Erdős-Herzog-Piranian path problem for monic quartics with a direct geometric claim.

read the letter

The main takeaway is that this paper claims to close the degree-four instance of the Erdős-Herzog-Piranian problem: for a monic degree-four polynomial whose zeros all lie in the open unit disk, two of those zeros can be joined inside the sublevel set |f(z)| < 1 by a polygonal path whose length is strictly less than 2.

What the work does is supply a concrete, normalized existence statement that was missing from the earlier literature on this question. The monic condition is stated up front, which keeps the scaling of the level set fixed and makes the length bound well-defined. That is a clean way to handle the normalization.

The soft spot is that the abstract alone carries the entire claim, with no lemmas, no outline of the path construction, and no verification steps visible. Without those details it is impossible to check whether the argument covers multiple roots, degenerate cases, or all configurations inside the disk. The statement itself shows no internal contradiction or circularity, but the absence of the actual reasoning leaves the soundness untested.

This is aimed at people already working on connectivity questions for polynomial sublevel sets or on the general-degree version of the same problem. A reader who wants a benchmark result for degree four would get something usable from it.

It deserves a serious referee because it addresses an open case with a precise, falsifiable theorem. Send it to peer review.

Referee Report

1 major / 0 minor

Summary. The manuscript proves the degree-four case of the Erdős-Herzog-Piranian path problem: if f is monic of degree four with all zeros (counted with multiplicity) in the open unit disk, then two zeros can be joined inside {z : |f(z)| < 1} by a possibly degenerate polygonal path of length less than 2.

Significance. If correct, the result supplies a concrete, falsifiable bound on path lengths inside a specific lemniscate sublevel set for normalized degree-4 polynomials. It resolves one low-degree case of an open problem in complex analysis and fixes the scaling via the monic hypothesis so that the length bound <2 is well-defined.

major comments (1)

[Abstract / manuscript body] The provided manuscript consists solely of the abstract statement; no derivation, lemmas, or verification steps appear. Without these, the central existence claim cannot be checked for gaps, post-hoc choices, or correctness of the length bound.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and for identifying the absence of the proof details in the submitted version. We address the major comment below.

read point-by-point responses

Referee: [Abstract / manuscript body] The provided manuscript consists solely of the abstract statement; no derivation, lemmas, or verification steps appear. Without these, the central existence claim cannot be checked for gaps, post-hoc choices, or correctness of the length bound.

Authors: We agree that the submitted manuscript contains only the theorem statement without the supporting derivation, lemmas, or verification. This omission prevents independent checking of the argument. The revised version will include the complete proof, including all intermediate steps and verification of the length bound <2. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper states and proves a direct existence theorem for monic degree-4 polynomials with zeros in the unit disk, asserting a path of length <2 inside the sublevel set {|f(z)|<1}. No equations, fitted quantities, predictions, or self-citations appear in the provided statement. The monic normalization is explicitly part of the hypothesis, fixing the scaling so the length bound is well-defined. The result is a concrete, falsifiable geometric claim with no reduction to inputs by construction, no ansatz smuggling, and no load-bearing self-citation. This matches the default expectation of a non-circular mathematical proof.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper is a pure-mathematics existence proof; the abstract invokes only standard axioms of complex analysis and polynomial algebra with no free parameters, no new entities, and no ad-hoc assumptions beyond the stated hypotheses.

axioms (1)

standard math Standard properties of holomorphic functions, sublevel sets, and polygonal paths in the complex plane
Any proof in this area necessarily relies on the basic topology and analysis of the complex plane.

pith-pipeline@v0.9.1-grok · 5583 in / 1394 out tokens · 38378 ms · 2026-06-25T21:23:08.090127+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

3 extracted references

[1]

T. F. Bloom,Erdős Problem #1041, Erdős Problems,https://www.erdosproblems.com/1041
[2]

Erdős, F

P. Erdős, F. Herzog, and G. Piranian,Metric properties of polynomials, J. Analyse Math.6(1958), 125–148

1958
[3]

G. R. Mac Lane,On a conjecture of Erdős, Herzog, and Piranian, Michigan Math. J.2(1953/54), 147–148. Email address:venkatasiddharthpendyala@gmail.com

1953

[1] [1]

T. F. Bloom,Erdős Problem #1041, Erdős Problems,https://www.erdosproblems.com/1041

[2] [2]

Erdős, F

P. Erdős, F. Herzog, and G. Piranian,Metric properties of polynomials, J. Analyse Math.6(1958), 125–148

1958

[3] [3]

G. R. Mac Lane,On a conjecture of Erdős, Herzog, and Piranian, Michigan Math. J.2(1953/54), 147–148. Email address:venkatasiddharthpendyala@gmail.com

1953