Under certain natural sufficient conditions on the sequence of uniformly bounded closed sets $E_k\subset\mathbb{ R}$ of admissible coefficients, we construct a polynomial $P_n(x)=1+\sum_{ k=1}^n\varepsilon_k x^k$, $\varepsilon_k\in E_k$, with at least $c\sqrt{ n}$ distinct roots in $[0,1]$, which matches the classical upper bound up to the value of the constant $c>0$. Our sufficient conditions cover the Littlewood ($E_k=\{ -1,1\}$) and Newman ($E_k=\{ 0,(-1)^k\}$) polynomials and are also necessary for the existence of such polynomials with arbitrarily many roots in the case when the sequence $E_k$ is periodic.
Categories: math.CA