On random polynomials with an intermediate number of real roots.

For each \alpha \in (0, 1), we construct a bounded monotone deterministic sequence (c_k)_{k \geqslant 0} of real numbers so that the number of real roots of the random polynomial f_n(z) = \sum _{k=0}^n c_k \varepsilon _k z^k is n^{\alpha + o(1)} with probability tending to one as the degree n tends to infinity, where (\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 (\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, including the asymptotic behavior of the variance and a central limit theorem. [ABSTRACT FROM AUTHOR]