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

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Aguirre, Ander , Nguyen, Hoi H. , Wang Jingheng · YOU? · · 2025 · Open Access ·

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.

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http://arxiv.org/abs/2511.07735

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https://arxiv.org/pdf/2511.07735

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

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2025

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2025-11-11

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Aguirre, Ander, Nguyen, Hoi H., Wang Jingheng

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https://arxiv.org/abs/2511.07735

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Mathematics, Term (time), Orthogonal polynomials, Interval (graph theory), Trigonometry, Zero (linguistics), Degree (music), Variance (accounting), Random variable, Random matrix, Pure mathematics, Classical orthogonal polynomials, Difference polynomials, Random walk, Trigonometric polynomial, Trigonometric functions, Discrete orthogonal polynomials, Discrete mathematics, Laguerre polynomials, Type (biology), Asymptotic formula, Mathematical analysis, Expected value, Combinatorics, Bernoulli polynomials, Complex quadratic polynomial, Wilson polynomials, Stochastic process, Mehler–Heine formula, Gegenbauer polynomials, Jacobi polynomials, Applied mathematics, Total variation, Moment (physics), Chebyshev polynomials, Asymptotic expansion, Random function

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