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arxiv: 2509.22035 · v2 · pith:QFTIHHA7new · submitted 2025-09-26 · 🧮 math.CV

Point evaluation for polynomials on the circle

Sarah May Instanes This is my paper

Pith reviewed 2026-05-18 13:01 UTC · model grok-4.3

classification 🧮 math.CV
keywords polynomialsunit circleL^p normssupremum normextremal constantsnorm inequalitiescomplex analysis
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The pith

The constant bounding the infinity norm of degree-d polynomials by their L^p norm on the unit circle is at most dp/2 + 1 in verified regimes.

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

The paper examines the smallest constant C such that the p-th power of the maximum value of any degree-d polynomial on the unit circle stays below C times the integral of the p-th power. This constant C_{d,p} measures the worst-case blow-up from average p-power to peak value. The authors conjecture that C_{d,p} never exceeds dp/2 plus one for p at least 2. They establish the bound for every p at least 2 when the degree is at most 4, and for every degree when p reaches 6.8 or higher. A reader would care because the result supplies an explicit linear control on pointwise size from an integral norm for all polynomials on the circle.

Core claim

The paper studies the constant C_{d,p} defined as the smallest constant C such that ||P||_∞^p ≤ C ||P||_p^p holds for every polynomial P of degree d, where the norms are taken on the unit circle. It conjectures that C_{d,p} ≤ dp/2 + 1 for all p ≥ 2 and all degrees d. The conjecture is shown to hold for all p ≥ 2 when d ≤ 4 and for all d when p ≥ 6.8.

What carries the argument

The extremal constant C_{d,p} given by the supremum of ||P||_∞^p / ||P||_p^p over all degree-d polynomials on the unit circle.

If this is right

  • For every degree at most 4 the bound dp/2 + 1 holds for all p at least 2.
  • For every p at least 6.8 the bound dp/2 + 1 holds for polynomials of any degree.
  • Explicit constructions and analytic estimates suffice to control the ratio in the proved regimes.
  • The controlling constant grows linearly in both degree and the exponent p within the verified ranges.

Where Pith is reading between the lines

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

  • Numerical search for degree-5 polynomials at moderate p values could test whether the linear bound is already sharp.
  • The same ratio bound may apply to trigonometric polynomials or to other closed curves in the plane.
  • As p tends to infinity the conjecture would recover a known linear growth of the maximum norm with degree.
  • Similar linear bounds might be testable for polynomials on the real line or in higher dimensions.

Load-bearing premise

The proofs for the verified regimes rest on the assumption that the extremal ratio is attained or bounded by explicit constructions or analytic estimates that do not miss larger values for the polynomials considered.

What would settle it

A polynomial of degree 5 at p=3 whose ratio ||P||_∞^p / ||P||_p^p exceeds 8.5 would falsify the conjecture in that regime.

Figures

Figures reproduced from arXiv: 2509.22035 by Sarah May Instanes.

Figure 1
Figure 1. Figure 1: The upper bound for Cd,p/(dp/2 + 1) for d = 2, d = 3 and d = 4 from Theorem 14. where Pd consists of all polynomials of degree at most d. For 1 < p < ∞, the extremal problem (1) has a unique solution φd,p. The extremal function φd,p is of degree d and has d zeroes on the unit circle T and P(e iθ) = P(e−iθ). Brevig, Chirre, Ortega-Cerdà and Seip [1] have studied the related problem of determining Cp defined… view at source ↗
Figure 2
Figure 2. Figure 2: The lower and upper bound for C4,p/(2p + 1). The lower bound is obtained by numerical optimization, while the upper bound is from Theorem 14. 2. Preliminaries We start with some preliminary results on the existence and structure of the solution of (1). Lemma 3. Let 1 < p < ∞. Then there exists a unique solution of (1). Proof. Let (Pn)n be a sequence in Pd such that Pn(1) = 1 and (∥Pn∥ p p )n converges to 1… view at source ↗
Figure 3
Figure 3. Figure 3: The curve Γε in the proof of Theorem 8 and the unit circle. The d zeroes of the polynomial are on the unit circle. Thus by a substitution and using the symmetry of the zeroes it follows that 1 π Z t1/2 0 fq((1 + ε)e iθ) (1 + ε)e iθ − 1 (1 + ε)e iθ + fq((1 + ε)e −iθ) (1 + ε)e−iθ − 1 (1 + ε)e −iθ dθ + 1 π Z π t1/2 fq((1 − ε)e iθ) (1 − ε)e iθ − 1 (1 − ε)e iθ + fq((1 − ε)e −iθ) (1 − ε)e−iθ − 1 (1 − ε)e −iθ dθ … view at source ↗
Figure 4
Figure 4. Figure 4: An illustration of the intervals at level 0 and level 1 for d = 3 and p = 10/3. Theorem 14 tells us that the supremum in equation (7) is attained whenever π/3 ≤ τ1 ≤ 5π/9. The shaded area represents Ep(τ ) without considering the integrand factor 1/ sin2 (θ/2). 1 9 π 2 9 π 1 3 π 5 9 π π [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The shaded area represents Ep(γ) for γ1 = 3π/18 with￾out considering the integrand factor 1/ sin2 (θ/2) for d = 3 and p = 10/3. Keep in mind that π/9 is the midpoint between the first interval at level 0 and the first interval at level 1. Lemma 12 tells us that for any γ1 in [π/9, π/3] it will follow that Ep(γ) ≤ Ep(τ ) where τ1 = π/3 as seen in [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Together with [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Together with [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: The expected zeroes of the extremal function φd,p for d = 6 and 2 ≤ p ≤ 4 based on numerical optimization [PITH_FULL_IMAGE:figures/full_fig_p015_8.png] view at source ↗
read the original abstract

We study the constant $\mathscr{C}_{d,p}$ defined as the smallest constant $C$ such that $\|P\|_\infty^p \leq C\|P\|_p^p$ holds for every polynomial $P$ of degree $d$, where we consider the $L^p$ norm on the unit circle. We conjecture that $\mathscr{C}_{d,p} \leq dp/2+1$ for all $p \geq 2$ and all degrees $d$. We show that the conjecture holds for all $p \geq 2$ when $d \leq 4$ and for all $d$ when $p \geq 6.8$.

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 / 2 minor

Summary. The paper defines the constant C_{d,p} as the smallest C such that ||P||_∞^p ≤ C ||P||_p^p for every polynomial P of degree d on the unit circle. It conjectures that C_{d,p} ≤ d p/2 + 1 for all p ≥ 2 and all d, and proves the conjecture holds for all p ≥ 2 when d ≤ 4 and for all d when p ≥ 6.8.

Significance. If the conjecture holds, the bound would control the pointwise evaluation of degree-d polynomials via their L^p norm on the circle, with potential uses in extremal problems and approximation theory. The manuscript's value is in the direct proofs for the regimes d ≤ 4 (all p ≥ 2) and p ≥ 6.8 (all d), which rely on explicit constructions and analytic estimates to verify the inequality in those cases.

minor comments (2)
  1. [Abstract] The abstract and introduction could include a short table or diagram summarizing the verified regimes (d ≤ 4 and p ≥ 6.8) to improve readability.
  2. [§1] Clarify the precise definition of the L^p norm on the unit circle (e.g., with respect to normalized Lebesgue measure) at the first appearance in §1.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive report and recommendation of minor revision. We appreciate the recognition of the potential uses in extremal problems and approximation theory. The referee's summary of the results is accurate.

Circularity Check

0 steps flagged

No circularity: direct proofs for special cases of the norm inequality conjecture

full rationale

The paper defines the constant C_{d,p} as the supremum of ||P||_∞^p / ||P||_p^p over degree-d polynomials on the unit circle and conjectures an upper bound of dp/2 + 1. It then proves the conjecture holds in the regimes d ≤ 4 (for all p ≥ 2) and p ≥ 6.8 (for all d) via explicit constructions and analytic estimates. No step reduces a claimed result to a fitted parameter, self-definition, or load-bearing self-citation; the arguments are self-contained mathematical derivations that do not loop back to their own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies entirely on standard axioms from complex analysis and functional analysis; no free parameters, new entities, or ad-hoc assumptions beyond the problem definition are introduced.

axioms (2)
  • domain assumption The unit circle is equipped with normalized Lebesgue measure for defining the L^p norm.
    This is the conventional setting for polynomial inequalities on the circle in complex analysis.
  • standard math Polynomials are elements of the space of holomorphic functions of degree at most d.
    Basic definition used throughout the statement of the inequality.

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Reference graph

Works this paper leans on

6 extracted references · 6 canonical work pages

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