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

Let Pₙ(x) = ∑ₖ₌₀ⁿ ξₖ xᵏ be a Kac random polynomial, where the coefficients ξₖ are i.i.d.\ copies of a given random variable $ξ$. Based on numerical experiments, it has been conjectured that if $ξ$ has mean zero, unit variance, and a finite (2+ε₀)-moment for some ε₀>0, then \[ E[NR(P_n)] \;=\; {2}π log n + C_ξ + o_n(1), \] where NR(Pₙ) denotes the number of real roots of Pₙ, and C_ξ is an absolute constant depending only on $ξ$, which is nonuniversal. Prior to this work, the existence of C_ξ had only been established by Do-Nguyen-Vu (2015, Proc.\ Lond.\ Math.\ Soc.) under the additional assumption that $ξ$ either admits a $(1+p)$-integrable density or is uniformly distributed on \± 1, ± 2, , ± N\. In this paper, using a different method, we remove these extra conditions on $ξ$, and extend the result to the setting where the ξₖ are independent but not necessarily identically distributed. Moreover, this proof strategy provides an alternative description of the constant C_ξ, and this new perspective serves as the key ingredient in establishing that C_ξ depends continuously on the distribution of $ξ$.