On random polynomials with an intermediate number of real roots

Abstract:

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.