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

In this paper, we investigate the number of real zeros of random Weyl polynomials with general coefficient distributions. Motivated by the results of [12, 25] as well as [6, 14], we determine how the expected number of real roots and their variance 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 term depends on the third and fourth moments of the distribution. In contrast, for the variance we show that the leading-order term is universal, differing from the behavior observed for random trigonometric polynomials in [6, 14]. This work represents a step toward establishing a central limit theorem for the fluctuations of the number of real roots. Our approach relies on an Edgeworth expansion for random walks arising from Weyl polynomials, a result of independent interest.