Improved lower bound for the number of unimodular zeros of self-reciprocal polynomials with coefficients in a finite set

Abstract

Let $n_1<n_2<\cdots <n_N$ be non-negative integers. In a private communication Brian Conrey asked how fast the number of real zeros of the trigonometric polynomials $T_N(\theta )={\sum }_{j=1}^N\cos (n_j\theta )$ tends to $\infty$ as a function of $N$. Conrey's question in general does not appear to be easy.Let ${\mathcal{P}}_n(S)$ be the set of all algebraic polynomials of degree at most $n$ with each of their coefficients in $S$. For a finite set $S\subset \mathbb{C}$ let $M=M(S):=\max \{|z|:z\in S\}$. It has been shown recently that if $S\subset \mathbb{R}$ is a finite set and $(P_n)$ is a sequence of self-reciprocal polynomials $P_n\in {\mathcal{P}}_n(S)$ with $|P_n(1)|$ tending to $\infty$, then the number of zeros of $P_n$ on the unit circle also tends to $\infty$. In this paper we show that if $S\subset \mathbb{Z}$ is a finite set, then every self-reciprocal polynomial $P\in {\mathcal{P}}_n(S)$ has at least $$c(\log \log \log |P(1)|{)}^{1-\epsilon }-1$$ zeros on the unit circle of $\mathbb{C}$ with a constant $c>0$ depending only on $\epsilon >0$ and$M=M(S)$. Our new result improves the exponent $1/2-\epsilon$ in a recent result by Julian Sahasrabudhe to $1-\epsilon$. Sahasrabudhe's new idea [66] is combined with the approach used in [34] offering an essencially simplified way to achieve our improvement. We note that in both Sahasrabudhe's paper and our paper the assumption that the finite set $S$ contains only integers is deeply exploited.

arXiv commentarXiv admin note: substantial text overlap with arXiv:1602.02284