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For each $\alpha \in (0, 1)$, we construct a bounded monotone deterministicsequence $(c_k)_{k \geq 0}$ of real numbers so that the number of real roots ofthe 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 toinfinity, where $(\varepsilon_k)$ is a sequence of i.i.d. (real) randomvariables of finite mean satisfying a mild anti-concentration assumption. Inparticular, this includes the case when $(\varepsilon_k)$ is a sequence ofi.i.d. standard Gaussian or Rademacher random variables. This result confirms aconjecture of O. Nguyen from 2019. More generally, our main results alsodescribe several statistical properties for the number of real roots of $f_n$,including the asymptotic behavior of the variance and a central limit theorem.

On random polynomials with an intermediate number of real roots