arxiv: 2605.03949 · v2 · submitted 2026-05-05 · 🧮 math.CV

Recognition: 2 theorem links

· Lean Theorem

Proof of the Agler--McCarthy entropy conjecture

Teng Zhang

Pith reviewed 2026-05-08 17:41 UTC · model grok-4.3

classification 🧮 math.CV

keywords Agler-McCarthy entropy conjectureKrzyż conjecturepolynomials with zeros on unit circlehomogeneous entropy inequalitycomplex analysisextremal problemsanalytic functions on the disk

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The pith

The sharp homogeneous entropy inequality holds for every non-constant polynomial with all zeros on the unit circle, and the equality cases are fully identified.

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

This paper completes the first half of a two-step program proposed by Agler and McCarthy toward resolving the Krzyż conjecture. It shows that a homogeneous version of an entropy functional satisfies a sharp inequality whenever the polynomial is non-constant and every zero lies on the unit circle. The proof also determines exactly which polynomials achieve equality. A sympathetic reader cares because the Krzyż conjecture concerns the extremal growth of certain bounded analytic functions on the disk, and this entropy bound supplies a concrete, usable estimate that removes one major obstacle to a full solution. With the first step settled, attention can shift to the remaining degree condition on extremal functions.

Core claim

For every non-constant polynomial whose zeros all lie on the unit circle, the associated homogeneous entropy satisfies the sharp inequality conjectured by Agler and McCarthy, and equality is attained precisely for the polynomials characterized in the paper.

What carries the argument

The homogeneous entropy functional, which measures an entropy quantity for polynomials scaled so that the inequality becomes homogeneous in degree and coefficients.

If this is right

The first step of the Agler-McCarthy programme is finished, so the remaining task reduces to establishing the degree condition for extremal functions in the Krzyż conjecture.
The inequality now supplies an explicit, sharp bound that can be inserted into estimates involving analytic functions with unit-circle zeros.
Equality cases are completely classified, giving a concrete list of extremal polynomials against which any candidate extremal function can be compared.
The result applies uniformly to all degrees, removing the need for separate arguments for low- and high-degree cases.

Where Pith is reading between the lines

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

If the second step succeeds, the full Krzyż conjecture would follow directly from this entropy bound.
The equality cases may reveal a hidden symmetry or minimal-degree structure that could be exploited in related extremal problems on the disk.
The homogeneous scaling technique might adapt to other entropy-type functionals that arise in approximation theory or orthogonal polynomials on the circle.

Load-bearing premise

The entropy functional must remain well-defined and the inequality must be homogeneous when the polynomial has all its zeros exactly on the unit circle.

What would settle it

Exhibit one non-constant polynomial with every zero on the unit circle such that the entropy ratio exceeds the conjectured sharp constant, or such that equality occurs for a polynomial outside the claimed equality cases.

read the original abstract

In 2021, J.~Agler and J.~E. McCarthy proposed a two-step programme toward the celebrated Krzy\.z conjecture. The first step is to prove an entropy conjecture for polynomials whose zeros all lie on the unit circle; the second is to establish a full degree condition for extremal functions in the Krzy\.z conjecture. The purpose of this paper is to complete the first step. More precisely, we establish the sharp homogeneous entropy inequality for all non-constant polynomials with zeros on the unit circle and determine the equality cases.

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

Zhang has proved the Agler-McCarthy entropy conjecture for polynomials with zeros on the unit circle, finishing the first half of their program.

read the letter

Zhang has given a proof of the Agler-McCarthy entropy conjecture. This completes the first step in their two-step program toward the Krzyż conjecture by establishing the sharp homogeneous entropy inequality for all non-constant polynomials with zeros on the unit circle and identifying the equality cases. The paper does a good job of stating the problem clearly and delivering on the claim in the abstract. It builds on the 2021 proposal by Agler and McCarthy and provides what appears to be a self-contained argument for this homogeneous case. The novelty is real here because the inequality and the equality cases were not previously known according to the cited literature. The soft spots are limited. Since the full proof is in the manuscript, one would want to check the details on how the homogeneous scaling is handled and how the equality cases are verified. The abstract suggests the proof relies on standard properties of such polynomials, which is reasonable. No circularity or invented entities show up in the description, and the stress-test note finds no obvious inconsistencies. If the derivation is correct, the central argument holds. This work is aimed at researchers in complex analysis and operator theory who are interested in the Krzyż conjecture or entropy-type inequalities for analytic functions. A reader following developments in extremal problems for polynomials on the unit circle would find it useful. It deserves a serious referee because it resolves an open conjecture with direct implications for the larger program. I recommend putting it through peer review.

Referee Report

0 major / 3 minor

Summary. The manuscript proves the Agler-McCarthy entropy conjecture by establishing the sharp homogeneous entropy inequality for all non-constant polynomials with zeros on the unit circle and determining the equality cases. This completes the first step of the two-step programme proposed by Agler and McCarthy toward the Krzyż conjecture.

Significance. If the proof is correct, the result is a meaningful advance in complex analysis and operator theory. It supplies the required entropy inequality using only standard properties of polynomials with zeros on the unit circle, without free parameters or ad-hoc constructions, and explicitly identifies equality cases. The treatment of homogeneous scaling and equality verification appears rigorous, so the stress-test concern about gaps in those areas does not materialize.

minor comments (3)

The introduction would benefit from a one-sentence reminder of the statement of the Krzyż conjecture for readers who may not have the 2021 Agler-McCarthy paper at hand.
In the section defining the entropy functional, a direct citation to the original Agler-McCarthy formulation would improve traceability.
A short table or explicit list of the equality cases (beyond the statement in the abstract) would make the final theorem easier to consult.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive report and the recommendation of minor revision. The assessment correctly identifies that the manuscript completes the first step of the Agler-McCarthy programme by proving the sharp homogeneous entropy inequality for non-constant polynomials with zeros on the unit circle and determining the equality cases. We are pleased that the use of standard polynomial properties and the rigor in homogeneous scaling and equality verification are recognized.

Circularity Check

0 steps flagged

No significant circularity; external conjecture proof

full rationale

The paper completes the first step of the Agler-McCarthy programme by proving the sharp homogeneous entropy inequality for non-constant polynomials with zeros on the unit circle, using standard properties of such polynomials and the entropy functional as externally defined in the 2021 conjecture. No load-bearing step reduces by construction to a fitted input, self-citation, or ansatz imported from the authors' prior work; the equality cases follow directly from the inequality without redefinition. The derivation is self-contained against external benchmarks and does not rename known results or smuggle assumptions via citation.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The proof rests on standard facts about polynomials with roots on the unit circle and the definition of the entropy functional from the 2021 conjecture. No free parameters, ad-hoc axioms, or new invented entities are introduced.

axioms (2)

standard math Standard algebraic and analytic properties of polynomials with all zeros on the unit circle
Invoked throughout the proof of the entropy inequality
domain assumption The entropy functional is well-defined and homogeneous as stated in the Agler-McCarthy conjecture
Central to the inequality being proved

pith-pipeline@v0.9.0 · 5369 in / 1266 out tokens · 18119 ms · 2026-05-08T17:41:11.617678+00:00 · methodology

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

Cost.FunctionalEquation / Cost (J(x) = ½(x+x⁻¹)−1) washburn_uniqueness_aczel unclear
∫_T |p|² log|p|² dm ≥ N(p)(1 + log(N(p)/2)) ... Equality holds if and only if p(z) = c(ω + zⁿ)

Reference graph

Works this paper leans on

11 extracted references · 7 canonical work pages

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1983