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

Let Pₙ (x) = ₊=₀^n 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+₀) -moment for some ₀>0, then \ E[Nₑ (Pₙ) \;=\; 2π n + C_ξ + oₙ (1), \] where Nₑ (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.