Improved lower bound for the number of unimodular zeros of self-reciprocal polynomials with coefficients in a finite set
Volume 192 / 2020
Abstract
Let $n_1 \lt n_2 \lt \cdots \lt n_N$ be nonnegative 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 $N$. This 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 $$c(\log \log \log |P(1)|)^{1-\varepsilon }-1$$ zeros on the unit circle of ${\mathbb C}$ with a constant $c \gt 0$ depending only on $\varepsilon \gt 0$ and $M = M(S)$. Our new result improves the exponent $1/2 - \varepsilon $ in a recent result by Sahasrabudhe to $1 - \varepsilon $. His new idea [Adv. Math. 343 (2019)] is combined with the approach used in our ealier work [Acta Arith. 176 (2016)] offering an essentially 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.
