On random polynomials with an intermediate number of real roots

Abstract

For each α ∈ ( 0 , 1 ) \alpha \in (0, 1) , we construct a bounded monotone deterministic sequence ( c k ) k ⩾ 0 (c_k)_{k \geqslant 0} of real numbers so that the number of real roots of the random polynomial f n ( z ) = ∑ k = 0 n c k ε k z k f_n(z) = \sum _{k=0}^n c_k \varepsilon _k z^k is n α + o ( 1 ) n^{\alpha + o(1)} with probability tending to one as the degree n n tends to infinity, where ( ε k ) (\varepsilon _k) is a sequence of i.i.d. (real) random variables of finite mean satisfying a mild anti-concentration assumption. In particular, this includes the case when ( ε k ) (\varepsilon _k) is a sequence of i.i.d. standard Gaussian or Rademacher random variables. This result confirms a conjecture of O. Nguyen from 2019. More generally, our main results also describe several statistical properties for the number of real roots of f n f_n , including the asymptotic behavior of the variance and a central limit theorem.