Central limit theorem for random polynomials with general coefficients

Let $N_{\mathbb{R},\boldsymbol{\xi}}$ be the number of real zeros of a random polynomial from any model considered in Section. Assume that the coefficient distribution is subgaussian and that $\xi$ has mean zero and variance one. Central limit theorem for general coefficients. For all such random polynomial models,

$$\frac{N_{\mathbb{R},\boldsymbol{\xi}}-\mathbb{E}N_{\mathbb{R},\boldsymbol{\xi}}}{\sqrt{\mathrm{Var}\,N_{\mathbb{R},\boldsymbol{\xi}}}} \xrightarrow{d}{\mathbf N}(0,1) \qquad\text{as }n\to\infty.$$

This conjecture proposes that the Gaussian fluctuation behavior known for several Gaussian polynomial models persists for subgaussian, non-Gaussian coefficient distributions. Its scope is all models considered in the paper, while the statement does not claim a proof for any particular model.