The aim of this paper is to further explore the number Nn(a, b) of real zeros of elliptic polynomials of degree n on any interval (a, b), where a and b may depend on n. We first obtain an asymptotic series for the variance of Nn(a, b) and the leading terms of the cumulants and central moments of Nn(a, b) in the large n limit. These terms play an important role in understanding the limiting law as well as the large n behavior of Nn(a, b). As a consequence, we then examine conditions on the interval (a, b) under which Nn(a, b) satisfies a central limit theorem and a strong law of large numbers.

We compute the precise leading asymptotics of the variance of the number of real roots for random polynomials whose coefficients have polynomial asymptotics. This class of random polynomials contains as special cases the Kac polynomials, hyperbolic polynomials, and any linear combinations of their derivatives. Prior to this paper, such asymptotics was only established in the 1970s for the Kac polynomials, with the seminal contribution of Maslova. The main ingredients of the proof are new asymptotic estimates for the two-point correlation function of the real roots, revealing geometric structures in the distribution of the real roots of these random polynomials. As a corollary, we obtain asymptotic normality for the number of real roots for these random polynomials, extending and strengthening a related result of O. Nguyen and V. Vu.