On random polynomials with an intermediate number of real roots

For each α ∈ (0, 1) (0, 1), we construct a bounded monotone deterministic sequence (c k) k ⩾ 0 (cₖ) ₊ 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 fₙ (z) = ₊=₀ⁿ cₖ ₖ zᵏ is n α + o (1) n^ + o (1) with probability tending to one as the degree n n tends to infinity, where (ε k) (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) (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 fₙ, including the asymptotic behavior of the variance and a central limit theorem.