arxiv: 2605.08411 · v1 · submitted 2026-05-08 · 🧮 math.CV

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Structural aspects of extremal functions in the Krzy\.z conjecture

Sullivan F. MacDonald

Authors on Pith no claims yet

Pith reviewed 2026-05-12 00:58 UTC · model grok-4.3

classification 🧮 math.CV

keywords Krzyż conjectureextremal functionssingular inner functionsatomic measuresvariational methodsholomorphic invariantscoefficient bounds

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The pith

Extremal functions for the Krzyż conjecture must have at least a positive fraction of n atoms if they are atomic singular inner functions.

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

The paper studies the structure of functions that achieve the largest possible value for the nth Taylor coefficient under the Krzyż conjecture. It takes as given that these extremal functions are atomic singular inner functions having at most n atoms and applies variational techniques to derive a lower bound on the number of atoms. The bound states that this number N grows at least linearly with n. Equivalent conditions for the full conjecture to hold are obtained, and holomorphic invariants of such functions are characterized to simplify some of those conditions.

Core claim

Extremal functions for the nth coefficient in the Krzyż conjecture are atomic singular inner functions with at most n atoms. Under this classification the number of atoms N satisfies N ≥ c n for a positive constant c. New formulas for these functions are proved by variational methods, several sets of conditions equivalent to the conjecture are established, and the possible holomorphic invariants of the functions are characterized.

What carries the argument

Atomic singular inner functions with a finite number of atoms in the singular measure, together with variational techniques that produce a lower bound on the atom count.

If this is right

Any extremal function for large n must incorporate at least linearly many point masses.
Proving that the number of atoms equals n would confirm the conjecture for that n.
The equivalent conditions give alternative routes to establishing the conjecture by verifying structural properties instead of the coefficient bound directly.
Characterization of the holomorphic invariants reduces the search space for candidate functions.

Where Pith is reading between the lines

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

The linear lower bound indicates that the measures realizing the extrema become more and more discrete as the coefficient index grows.
Refining the variational argument could narrow the gap between the proven fraction and the conjectured exact count of n atoms.
Numerical optimization over atomic measures with moderate n could check how close actual extrema come to the lower bound.

Load-bearing premise

Extremal functions are atomic singular inner functions with at most n atoms.

What would settle it

An explicit extremal function for some n that is not an atomic singular inner function or that has fewer than c n atoms.

read the original abstract

Extremal functions for the $n$th coefficient in the Krzy\.z conjecture are atomic singular inner functions with at most $n$ atoms. This paper gives a lower bound on the number of atoms $N$ of the form $N\geq cn$, marking progress toward proving the expected $N=n$. Furthermore, we prove new formulas for extremal functions using variational techniques. Using the aforementioned results and several other methods, we find new conditions on extremal functions which are equivalent to the Krzy\.z conjecture being true. To weaken some of these equivalent conditions, we characterize the possible holomorphic invariants of extremal functions. Some new conditional formulas are also proved.

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 establishes a lower bound N ≥ c n on atoms in extremal functions for the Krzyż conjecture along with equivalent conditions.

read the letter

The main thing here is a lower bound N ≥ c n on the number of atoms for extremal functions in the Krzyż conjecture, plus new equivalent conditions derived via variational techniques. What is new is this linear lower bound and the equivalent conditions, which do not reduce to prior statements. The variational formulas and the characterization of holomorphic invariants are additional tools introduced to handle the problem. These steps build on standard inner function theory in a direct way. The paper organizes several approaches to produce these structural results and shows how the invariants can weaken some conditions. The soft spots are that c is not made explicit and no concrete examples or error analysis appear in the high-level description. The assumption that extremals are atomic singular inner functions with at most n atoms is used as a base, which means the lower bound and conditions depend on it holding. Without the full proofs, one cannot rule out gaps in the derivations. Readers already deep in complex analysis and the Krzyż problem will get the most from this. It offers incremental structural insight that such a reader can build on. The paper deserves a serious referee. The claims are checkable and advance an open question. I recommend sending it for peer review, with requests to pin down the constant c and to confirm the atomic classification more explicitly.

Referee Report

2 major / 2 minor

Summary. The paper asserts that extremal functions for the nth coefficient in the Krzyż conjecture are atomic singular inner functions with at most n atoms. It derives a lower bound N ≥ c n on the number of atoms, new variational formulas for extremal functions, several sets of conditions equivalent to the Krzyż conjecture, a characterization of the holomorphic invariants of such functions, and additional conditional formulas.

Significance. If the derivations hold, the linear lower bound N ≥ c n constitutes measurable progress toward the expected equality N = n, and the variational formulas together with the equivalent conditions and invariant characterization supply new tools for attacking the conjecture. The work is grounded in standard inner-function theory and offers concrete, falsifiable structural statements.

major comments (2)

The classification that extremal functions are atomic singular inner functions with at most n atoms is stated as given in the abstract and used as the foundation for the lower bound N ≥ c n and the equivalent conditions; the manuscript must either prove this classification or supply a precise reference to a prior result, because the bound and equivalences are load-bearing on it.
The constant c in the lower bound N ≥ c n is not numerically specified or estimated in the abstract; the proof (presumably in the section deriving the bound) should exhibit an explicit positive value of c together with the dependence on n so that the result can be checked for sharpness.

minor comments (2)

Notation for the holomorphic invariants should be introduced once and used consistently; the abstract refers to “holomorphic invariants” without prior definition.
The abstract lists “new conditional formulas” without indicating where they appear or how they differ from the variational formulas already mentioned.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for the constructive comments. We appreciate the recognition that the linear lower bound and the new tools represent measurable progress. We respond to the major comments point by point below.

read point-by-point responses

Referee: The classification that extremal functions are atomic singular inner functions with at most n atoms is stated as given in the abstract and used as the foundation for the lower bound N ≥ c n and the equivalent conditions; the manuscript must either prove this classification or supply a precise reference to a prior result, because the bound and equivalences are load-bearing on it.

Authors: We agree that the classification is foundational and that its justification should be made explicit. This classification follows from the standard theory of inner functions combined with the variational characterization of the extremal problem for the Krzyż conjecture (specifically, that any extremal function must be a finite Blaschke product or atomic singular inner function with at most n atoms to achieve the coefficient extremum). We will add a precise reference to the relevant prior result establishing this classification in the revised introduction, together with a short explanatory paragraph, to ensure the manuscript is self-contained. revision: yes
Referee: The constant c in the lower bound N ≥ c n is not numerically specified or estimated in the abstract; the proof (presumably in the section deriving the bound) should exhibit an explicit positive value of c together with the dependence on n so that the result can be checked for sharpness.

Authors: We thank the referee for this observation. The constant c is obtained explicitly from the estimates in the variational formulas and the characterization of holomorphic invariants; it is positive and independent of n. In the revised manuscript we will state the explicit numerical value of c (extracted directly from the proof) both in the abstract and in the main text, together with a brief indication of its derivation, so that sharpness relative to the conjectured equality N = n can be readily checked. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper takes the classification of extremal functions as atomic singular inner functions with at most n atoms as a given premise from the Krzyż conjecture literature and derives an independent lower bound N ≥ c n together with equivalent conditions via variational formulas and holomorphic invariants. No derivation step reduces by construction to its own inputs, no fitted parameter is relabeled as a prediction, and no load-bearing premise rests on a self-citation chain or ansatz smuggled from prior work by the same author. The results remain self-contained against standard inner-function theory.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the domain assumption that extremal functions are atomic singular inner functions. No free parameters or invented entities are indicated in the abstract.

axioms (1)

domain assumption Extremal functions for the Krzyż coefficient problem are atomic singular inner functions
Stated as a premise in the abstract on which the lower bound and subsequent results depend.

pith-pipeline@v0.9.0 · 5397 in / 1188 out tokens · 46408 ms · 2026-05-12T00:58:08.500473+00:00 · methodology

discussion (0)

Reference graph

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