On the First Non-Universal Term in Random Polynomial Real Zeros

Let $P_n(x)={\sum }_{k=0}^n{\xi }_k{x}^k$ be a Kac random polynomial, where the coefficients ${\xi }_k$ are i.i.d.\ copies of a given random variable $\xi$. Based on numerical experiments, it has been conjectured that if $\xi$ has mean zero, unit variance, and a finite $(2+{\epsilon }_0)$-moment for some ${\epsilon }_0>0$, then $$\mathbb{E}[N_{\mathbb{R}}(P_n)]=\frac{2}{\pi }\log n+C_{\xi }+o_n(1),$$ where $N_{\mathbb{R}}(P_n)$ denotes the number of real roots of $P_n$, and $C_{\xi }$ is an absolute constant depending only on $\xi$, which is nonuniversal. Prior to this work, the existence of $C_{\xi }$ had only been established by Do-Nguyen-Vu (2015, \emph{Proc.\ Lond.\ Math.\ Soc.}) under the additional assumption that $\xi$ either admits a $(1+p)$-integrable density or is uniformly distributed on $\{\pm 1,\pm 2,\dots ,\pm N\}$. In this paper, using a different method, we remove these extra conditions on $\xi$, and extend the result to the setting where the ${\xi }_k$ are independent but not necessarily identically distributed. Moreover, this proof strategy provides an alternative description of the constant $C_{\xi }$, and this new perspective serves as the key ingredient in establishing that $C_{\xi }$ depends continuously on the distribution of $\xi$.