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arxiv: 2605.17386 · v1 · pith:EEJLS36Gnew · submitted 2026-05-17 · 🧮 math.CV

On Average Modulus of Random Polynomials Over a Unit Circle and Disc

Sajad A. Sheikh , Mohd.Ibrahim Mir This is my paper

Pith reviewed 2026-05-19 22:41 UTC · model grok-4.3

classification 🧮 math.CV
keywords random polynomialsaverage modulusunit circleunit diskMarkov inequalitymaximum modulusGaussian coefficientsuniform coefficients
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The pith

Random polynomials with i.i.d. standard normal coefficients have their average modulus on the unit circle and disk newly characterized, with maximum-modulus tail probabilities bounded by Markov inequality.

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

The paper establishes new results for the expected modulus of random polynomials with coefficients drawn i.i.d. from the standard normal distribution, evaluated both on the unit circle and throughout the closed unit disk. These quantities describe the typical size of such polynomials in the complex plane and therefore matter for any study of random analytic functions or their norms. The authors further obtain explicit upper bounds on the probability that the maximum modulus exceeds a prescribed positive threshold, first for Gaussian coefficients and then for uniform coefficients, by direct application of the Markov inequality. The same probabilistic technique is presented as a starting point for a wider class of problems on norms of random polynomials.

Core claim

For a random polynomial whose coefficients are independent standard normal random variables, the average modulus on the unit circle and on the unit disk admits new characterizations; for both Gaussian and uniform coefficient distributions, Markov's inequality supplies an upper bound on the probability that the maximum modulus on these sets exceeds any fixed positive number.

What carries the argument

Markov's inequality applied to the random variable given by the maximum modulus, together with direct computation of the expected modulus under normal coefficients.

Load-bearing premise

The coefficients are independent and identically distributed as standard normal, Gaussian, or uniform random variables.

What would settle it

Direct evaluation of the average modulus for a concrete low-degree polynomial with standard normal coefficients that deviates from the paper's stated characterization, or a numerical check showing that the probability the maximum modulus exceeds the threshold lies above the Markov-derived bound.

read the original abstract

This article presents some interesting and novel results concerning the average modulus of random polynomials on the unit circle and the unit disc, with coefficients distributed as standard normal variates. The paper also introduces new results concerning the bounds of the maximum modulus of random polynomials with coefficients distributed as independently as Gaussian and uniform variates, utilizing probability principles to derive findings about the likelihood of the maximum modulus exceeding a specific threshold, using Markov inequality as the primary probabilistic tool. These findings and the approach can potentially initiate the study of a rich class of problems concerning the norms of random polynomials.

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

2 major / 2 minor

Summary. The manuscript derives expressions for the expected modulus of random polynomials with i.i.d. standard normal coefficients evaluated on the unit circle and within the closed unit disc. It also derives tail bounds on the maximum modulus for both Gaussian and uniform coefficients via direct application of Markov's inequality.

Significance. If the derivations hold, the explicit formulas for E[|p(z)|] would be a modest but useful addition to the literature on random polynomials, as they follow from elementary properties of Gaussians and geometric sums. The Markov bounds are elementary and could serve as a starting point for further tail estimates, consistent with the abstract's claim of initiating study of a rich class of problems.

major comments (2)
  1. [§2] §2 (or the section deriving the average modulus): the claimed closed-form E[|p(z)|] = sqrt(π σ²/2) with σ² = n+1 on the unit circle assumes a circularly symmetric complex Gaussian. With real-valued standard normal coefficients (as stated in the abstract), p(z) for |z|=1 is bivariate normal whose covariance matrix is generally anisotropic and depends on arg(z); at z=1 it reduces to a real Gaussian yielding E[|p(1)|] = sqrt(2(n+1)/π). This directly affects the central claim and must be corrected or the coefficient field clarified.
  2. [Markov bound section] The Markov bound section (likely §4): the argument requires an explicit dominating random variable (e.g., the L² norm on the circle or a finite discretization) whose expectation is finite and controlled uniformly in the degree. The manuscript must state this dominating variable and verify that the resulting bound on P(max |p| > t) is non-vacuous and matches the stated assumptions on the coefficients.
minor comments (2)
  1. [Abstract] Abstract: the phrase 'standard normal variates' should be expanded to 'real-valued' or 'complex-valued' to avoid ambiguity that propagates through the derivations.
  2. [Notation] Notation: ensure the degree is consistently denoted (e.g., polynomials of exact degree n versus sum up to n) and that the geometric sum for the variance inside the disc is written explicitly as (1−|z|^{2(n+1)})/(1−|z|^2).

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below and will make the indicated revisions to strengthen the paper.

read point-by-point responses

  1. Referee: [§2] §2 (or the section deriving the average modulus): the claimed closed-form E[|p(z)|] = sqrt(π σ²/2) with σ² = n+1 on the unit circle assumes a circularly symmetric complex Gaussian. With real-valued standard normal coefficients (as stated in the abstract), p(z) for |z|=1 is bivariate normal whose covariance matrix is generally anisotropic and depends on arg(z); at z=1 it reduces to a real Gaussian yielding E[|p(1)|] = sqrt(2(n+1)/π). This directly affects the central claim and must be corrected or the coefficient field clarified.

    Authors: We appreciate the referee's precise observation. The manuscript states that the coefficients are real-valued i.i.d. standard normals. The derivation in §2 incorrectly invoked the circular symmetry of complex Gaussians. We will correct this section by deriving the proper expression for E[|p(z)|] under real coefficients, where for |z|=1 the real and imaginary parts form a bivariate normal with covariance depending on arg(z). The revised formula will be stated explicitly, including the special case E[|p(1)|] = sqrt(2(n+1)/π). revision: yes

  2. Referee: [Markov bound section] The Markov bound section (likely §4): the argument requires an explicit dominating random variable (e.g., the L² norm on the circle or a finite discretization) whose expectation is finite and controlled uniformly in the degree. The manuscript must state this dominating variable and verify that the resulting bound on P(max |p| > t) is non-vacuous and matches the stated assumptions on the coefficients.

    Authors: We thank the referee for this important clarification. In the revised manuscript we will explicitly introduce a dominating random variable (for example, the L² norm of p on the unit circle or the maximum of |p| over a finite ε-net on the circle) and verify that its expectation remains finite and bounded uniformly in the degree n, for both Gaussian and uniform coefficient assumptions. This will ensure the resulting Markov bound on P(max |p| > t) is non-vacuous and rigorously justified. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivations are self-contained

full rationale

The paper computes expectations of |p(z)| for random polynomials with i.i.d. Gaussian coefficients by direct summation of variances (geometric series inside the disc, constant n+1 on the circle) and applies the standard Markov inequality to bound the maximum modulus. These steps invoke only elementary properties of centered Gaussians and the Markov inequality itself, both of which are external to the paper and not defined in terms of its own results. No self-citations, fitted parameters renamed as predictions, or ansatzes smuggled via prior work appear in the load-bearing derivations. The central claims therefore reduce to explicit closed-form expressions and standard tail bounds rather than to any input by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based solely on the abstract, no free parameters, axioms, or invented entities can be identified; the work appears to rest on standard assumptions that coefficients are i.i.d. normal or uniform and on the classical Markov inequality.

pith-pipeline@v0.9.0 · 5615 in / 1098 out tokens · 30806 ms · 2026-05-19T22:41:00.531208+00:00 · methodology

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