The number of unimodular zeros of self-reciprocal polynomials with coefficients in a finite set
Volume 176 / 2016
Abstract
Let ${\rm NZ}(T_n)$ denote the number of real zeros of a trigonometric polynomial $$T_n(t) = \sum_{j=0}^n{a_{j,n} \cos(jt)}, \ \quad a_{j,n} \in {\mathbb C},$$ in a period $[a,a+2\pi)$, $a \in {\mathbb R}$. Let ${\rm NZ}(P_n)$ denote the number of zeros of an algebraic polynomial $$P_n(z) = \sum_{j=0}^n{p_{j,n} z^j}, \ \quad p_{j,n} \in {\mathbb C},$$ that lie on the unit circle of ${\mathbb C}$. Let $$ {\rm NC}_k(P_n) := \Big|\Big\{u: 0 \leq u \leq n-k+1, \, \sum_{j=u}^{u+k-1}{p_{j,n}} \neq 0 \Big\}\Big|.$$ One of the highlights of this paper states that $\lim_{n \rightarrow \infty}{ {\rm NZ}(T_n)} = \infty$ whenever the set $\{a_{j,n}: j \in \{0,1,\ldots,n\}, \, n \in {\mathbb N}\} \subset [0,\infty)$ is finite and $$\lim_{n \rightarrow \infty}{|\{j \in \{0,1,\ldots,n\}:a_{j,n} \neq 0\}|} = \infty.$$ This follows from a more general result stating that $$\lim_{n \rightarrow \infty}{{\rm NZ}(P_{2n})} = \infty$$ whenever $P_{2n}$ is self-reciprocal, the set $\{p_{j,2n}: j \in \{0,1,\ldots,2n\}, \, n \in {\mathbb N}\} \subset {\mathbb R}$ is finite, and $\lim_{n \rightarrow \infty}{{\rm NC}_k(P_{2n})} = \infty$ for every $k \in {\mathbb N}$.
