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The Newman algorithm for constructing polynomials with restricted coefficients and many real roots (2404.07971v1)
Published 11 Apr 2024 in math.CA
Abstract: 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 $P_n(x)=1+\sum_{k=1}n\varepsilon_k xk$, $\varepsilon_k\in E_k$, 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.