Extensions of the Bloch -- Pólya Theorem on the number of real zeros of polynomials (II)

Tamás Erdélyi

Published 2024-07-23Version 1

We prove that there are absolute constants $c_1 > 0$ and $c_2 > 0$ such that for every $$a_0,a_1, \ldots,a_n \in [1,M]\,, \qquad 1 \leq M \leq \exp(c_2n)\,,$$ there are $$b_0,b_1,\ldots,b_n \in \{-1,0,1\}$$ such that the polynomial $\displaystyle{P(z) = \sum_{j=0}^n{b_ja_jz^j}}$ has at least $$c_1 \left( \frac{n}{\log(4M)} \right)^{1/2}$$ distinct sign changes in $I_a := (1-2a,1-a)$, where $\displaystyle{a := \left( \frac{\log(4M)}{n} \right)^{1/2} \leq 1/3}$. This improves and extends earlier results of Bloch and P\'olya and Erd\'elyi, and, as a special case, recaptures a special case of a more general recent result of Jacob and Nazarov.