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arxiv: 2605.08411 · v2 · pith:K2YISSWGnew · submitted 2026-05-08 · 🧮 math.CV

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

Sullivan F. MacDonald This is my paper

Pith reviewed 2026-05-20 22:23 UTC · model grok-4.3

classification 🧮 math.CV
keywords Krzyż conjectureextremal functionsatomic singular inner functionsvariational techniquesanalytic invariantscoefficient boundssingular inner functions
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The pith

Extremal functions for the Krzyż conjecture's nth coefficient are atomic singular inner functions with at least cn atoms.

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

The paper establishes that any function extremal for the nth coefficient in the Krzyż conjecture must be an atomic singular inner function with at least cn atoms for a positive constant c. This lower bound represents progress toward the expected exact count of n atoms. Variational techniques yield new formulas for these functions, and several analytic conditions on them are shown to be equivalent to the conjecture holding. A reader would care because these structural results narrow the search for the true maximizers and offer alternative routes to settling a classical coefficient problem in complex analysis.

Core claim

Extremal functions for the nth coefficient in the Krzyż conjecture are atomic singular inner functions with at most n atoms. The paper proves a lower bound N ≥ cn on the number of atoms and derives new formulas for these functions via variational techniques. It further establishes several new analytic conditions equivalent to the conjecture being true and characterizes the possible analytic invariants of the extremal functions.

What carries the argument

Atomic singular inner functions with finitely many atoms, which represent the candidate extremal functions for maximizing the nth coefficient.

If this is right

  • Extremal functions must have at least linearly many atoms in their representation.
  • Variational methods produce new explicit formulas for the extremal functions.
  • Several analytic conditions on extremal functions are equivalent to the Krzyż conjecture holding.
  • The analytic invariants of extremal functions admit a complete characterization.

Where Pith is reading between the lines

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

  • Strengthening the constant c to exactly 1 would imply that extremal functions have precisely n atoms and thereby prove the conjecture.
  • The equivalent conditions could support numerical searches for counterexamples at moderate n.
  • The invariants characterization may extend to related extremal problems for other classes of analytic functions.

Load-bearing premise

The extremal functions are precisely the atomic singular inner functions with finitely many atoms, and variational techniques apply to them without extra regularity conditions that might exclude the true maximizers.

What would settle it

An explicit example of an extremal function for some large n that has fewer than cn atoms would disprove the lower bound.

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 these results and several other methods, we establish new conditions on extremal functions which are equivalent to the Krzy\.z conjecture being true. We also characterize the possible analytic invariants of extremal functions.

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

1 major / 1 minor

Summary. The manuscript asserts that extremal functions for the nth coefficient in the Krzyż conjecture are atomic singular inner functions with at most n atoms. It establishes a lower bound N ≥ c n on the number of atoms for some positive constant c, derives new formulas for these functions via variational techniques, proves several new analytic conditions equivalent to the Krzyż conjecture holding, and characterizes possible analytic invariants of the extremal functions.

Significance. If the central derivations hold, the work advances understanding of the Krzyż conjecture by quantifying a linear lower bound on the complexity of extremal functions and supplying equivalent analytic conditions that could support further progress. The characterization of invariants adds structural information, and the variational formulas constitute a concrete technical contribution when rigorously justified.

major comments (1)
  1. The derivation of new formulas for extremal functions via variational techniques (appearing after the statement that extremal functions are atomic singular inner functions with at most n atoms) assumes that these functions admit sufficiently regular variations remaining inside the admissible class. Atomic singular inner functions with finitely many atoms possess jump discontinuities on the unit circle; it is not immediate that the first variation vanishes in the usual Gateaux or Euler-Lagrange sense without additional approximation arguments or a weak-sense formulation. This point is load-bearing because both the lower bound N ≥ c n and the equivalent analytic conditions rest directly on the validity of these formulas.
minor comments (1)
  1. The abstract states both an upper bound of n atoms and a lower bound of c n atoms; a brief remark on how these bounds relate and whether the upper bound is proved or assumed would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for identifying a point that merits explicit clarification. We address the major comment below and will incorporate a short explanatory paragraph in the revision to make the nature of the variations fully transparent.

read point-by-point responses

  1. Referee: The derivation of new formulas for extremal functions via variational techniques (appearing after the statement that extremal functions are atomic singular inner functions with at most n atoms) assumes that these functions admit sufficiently regular variations remaining inside the admissible class. Atomic singular inner functions with finitely many atoms possess jump discontinuities on the unit circle; it is not immediate that the first variation vanishes in the usual Gateaux or Euler-Lagrange sense without additional approximation arguments or a weak-sense formulation. This point is load-bearing because both the lower bound N ≥ c n and the equivalent analytic conditions rest directly on the validity of these formulas.

    Authors: We appreciate the referee drawing attention to this technical point. In the paper the variations are not taken in the full space of inner functions; they are parametric variations within the finite-dimensional family of atomic singular inner functions having a fixed number of atoms. Concretely, an extremal function is determined by the angular positions θ_k and the positive masses m_k of its atoms. A variation consists of replacing (θ_k, m_k) by (θ_k + ε δθ_k, m_k + ε δm_k) for small ε, with the total mass kept constant if required by the normalization. The resulting function remains an atomic singular inner function with the same number of atoms and therefore stays inside the admissible class. Because the map from the finite set of parameters to the nth Taylor coefficient is smooth (in fact real-analytic) on this parameter domain, the first variation vanishing is simply the statement that all partial derivatives with respect to the θ_k and m_k are zero at a critical point. This is ordinary multivariable calculus and does not invoke Gateaux differentiability in the infinite-dimensional function space, nor does it require the boundary function to be continuous. The jump discontinuities are therefore irrelevant to the differentiation step. The lower bound N ≥ c n and the equivalent analytic conditions are obtained by substituting these first-order conditions into the coefficient functional; the argument is therefore self-contained within the parametric setting. In the revised manuscript we will add a brief paragraph immediately after the statement that extremal functions are atomic, explicitly describing the parameter space and confirming that the first-order condition is the standard vanishing of the gradient in these coordinates. revision: partial

Circularity Check

0 steps flagged

No circularity: derivation relies on variational arguments and prior structural results independent of the target conjecture

full rationale

The paper states the extremal functions are atomic singular inner functions with at most n atoms as a starting point, then applies variational techniques to derive new formulas, a lower bound N ≥ cn, and equivalent analytic conditions. No quoted step reduces a prediction or central claim to a fitted parameter, self-citation chain, or definitional tautology by construction. The variational formulas and equivalence conditions are presented as derived outputs rather than inputs renamed, and the abstract gives no evidence of load-bearing self-citations or ansatz smuggling. The derivation chain remains self-contained against external benchmarks in the provided text.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on the classical theory of inner functions and variational methods in the unit disk. No new free parameters, invented entities, or ad-hoc axioms are introduced in the abstract.

axioms (1)
  • domain assumption Extremal functions for the Krzyż coefficient problem are atomic singular inner functions.
    Stated as the opening premise of the abstract; the entire analysis proceeds from this classification.

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