Real Roots of Random Weyl Polynomials with General Coefficients: Expectation and Variance

TL;DR

This work studies the real zeros of Weyl polynomials with general i.i.d. coefficients, revealing universal leading behavior for the number of real zeros while identifying non-universal, moment-dependent corrections at next order. The authors develop and deploy an Edgeworth expansion framework for linear forms, combining Kac-Rice formulas with precise cumulant-based corrections and high-dimensional Diophantine (LCD) conditions to compare general-coefficient models to the Gaussian baseline. They prove that the bulk expectation correction scales with a logarithm whose coefficient depends on the third and fourth moments of the coefficient distribution, while the variance’s leading term remains universal across natural intervals, with non-universal effects residing in lower-order terms. The analysis hinges on control of small-ball probabilities and a robust Edgeworth toolkit, offering a path toward refined fluctuation results and potential CLTs for Weyl polynomials and related ensembles. The results sharpen understanding of universality versus non-universality in random polynomial zeros and provide tools potentially extensible to broader factorial-type ensembles and edge regimes.

Abstract

In this paper, we investigate the number of real zeros of random Weyl polynomials of degree \(n \to \infty\) with general coefficient distributions. Motivated by the results of arXiv:1409.4128 and arXiv:1402.4628 as well as arXiv:1711.03316 and arXiv:1912.11901, we determine how the expected number of real zeros and their variance, over various natural intervals, depend on the moments of the common coefficient distribution. Our main finding is that while the first-order asymptotic of the expectation is universal, the next-order correction depends on the third and fourth moments of the distribution, and may grow linearly with \(\log n\), depending on the interval under consideration. In contrast, for the variance we show that the leading-order term is universal, which differs from the behavior observed for random trigonometric polynomials in arXiv:1711.03316 and arXiv:1912.11901. Our approach relies on an Edgeworth expansion for random walks arising from Weyl polynomials, a result of independent interest.

Theorems & Definitions (65)

• Theorem 1.1: Local universality of correlations
• Corollary 1.3: DONgVTV
• Theorem 1.4
• Theorem 1.6
• Remark 1.7
• Conjecture 1.8: Linearity of the variance
• Theorem 1.9
• Theorem 1.11: Main result: expectation for real zeros of Weyl polynomials
• Theorem 1.12: Main result: variance for real zeros of Weyl polynomials
• Theorem 2.2