AI 中文总结
本文证明了一般系数（对称次高斯分布）Weyl多项式实零点总数的中心极限定理，推广了高斯情形结果至非高斯分布，并处理了非对称情形下的区间零点计数。
AI 中文摘要
对于随机多项式，其实零点个数$N_{\mathbb R}$是其系数的高度非线性函数，其统计性质已被广泛研究。一个自然的问题是$N_{\mathbb R}$是否满足中心极限定理。对于具有独立同分布标准高斯系数的各种系综，此类中心极限定理已被建立；例如，参见arXiv:1911.12182、arXiv:1801.06331、arXiv:1401.5745；Azais和Leon，Electron. J. Probab. 18 (2013), no. 68；arXiv:1504.05355、arXiv:2111.09015、arXiv:1707.09276、arXiv:1005.4113。这些结果依赖于丰富的工具，包括Kac-Rice公式、矩方法和Wiener混沌分解。然而，在非高斯情形下，这些工具中的许多是不可用的。据我们所知，此前高斯情形之外的中心极限定理仅限于Kac型多项式，包括双曲多项式；参见Maslova (1974)、O. Nguyen和Vu (arXiv:1904.04347)的工作，以及更近期的Do、N. Nguyen和O'Rourke (arXiv:2605.26402)的工作。在本文中，我们证明了Weyl多项式的实零点总数满足中心极限定理，其中系数是某个对称、零均值、方差为一的次高斯随机变量$\xi$的独立同分布副本。这实质上将Do和Vu (arXiv:1707.09276)的主要结果之一推广到了包括Rademacher分布在内的广泛非高斯分布类。在不假设对称性的情况下，我们证明了正质量区间以及$[0,\infty)$上实零点个数的中心极限定理。我们的证明结合了我们近期工作(arXiv:2511.07735)中的均匀单点反集中估计与Weyl多项式在系数指标$i\approx x^2$附近的局部化性质。虽然我们的证明使用比较来计算方差，但中心极限定理的推导是相当直接的。
英文摘要
For a random polynomial, the number of real zeros $N_{\mathbb R}$ is a highly nonlinear function of its coefficients, and its statistical properties have been studied extensively. A natural question is whether $N_{\mathbb R}$ satisfies a central limit theorem. For various ensembles with iid standard Gaussian coefficients, such central limit theorems have been established; see, for instance, arXiv:1911.12182, arXiv:1801.06331, arXiv:1401.5745; Azais and Leon, Electron. J. Probab. 18 (2013), no. 68; arXiv:1504.05355, arXiv:2111.09015, arXiv:1707.09276, arXiv:1005.4113. These results rely on a rich range of tools, including Kac-Rice formulas, moment methods, and Wiener chaos decompositions. In the non-Gaussian setting, however, many of these tools are unavailable. To the best of our knowledge, prior central limit theorems beyond the Gaussian setting were limited to Kac-type polynomials, including hyperbolic polynomials; see the works of Maslova (1974), O. Nguyen and Vu (arXiv:1904.04347), and, more recently, Do, N. Nguyen, and O'Rourke (arXiv:2605.26402). In this paper, we prove a central limit theorem for the total number of real zeros of Weyl polynomials whose coefficients are iid copies of a symmetric, mean-zero, variance-one subgaussian random variable $ξ$. This substantially extends one of the main results of Do and Vu (arXiv:1707.09276) to a broad class of non-Gaussian distributions, including the Rademacher distribution. Without the symmetry assumption, we prove central limit theorems for the number of real zeros for positive bulk intervals, as well as for $[0,\infty)$. Our proof combines the uniform one-point anti-concentration estimates from our recent work (arXiv:2511.07735) with the localization of Weyl polynomials around the coefficient index $i\approx x^2$. While our proofs use comparison to compute the variances, the CLT deduction is rather direct.
