On the zeros of reciprocal Littlewood polynomials
Streszczenie
Let $P(z)=\sum_{n=0}^Na_nz^n$ be a Littlewood polynomial of degree $N$, meaning that $a_n\in \{-1, 1\}$. We say that $P$ is reciprocal if $P(z)=z^NP(1/z)$. Borwein, Erdélyi and Littmann (2008) posed the question of determining the minimum number $Z_{\mathcal {L}}(N)$ of zeros of modulus 1 of a reciprocal Littlewood polynomial $P$ of degree $N$. Several finite lower bounds on $Z_{\mathcal {L}}(N)$ have been obtained in the literature, and it has been conjectured by various authors that $Z_{\mathcal {L}}(N)$ must in fact grow to infinity with $N$. Starting from ideas in recent breakthrough papers of Erdélyi (2020) and Sahasrabudhe (2019), we are able to confirm this. In particular, this answers a question of Erdélyi by proving that the number of zeros of a cosine polynomial $$\sum _{n=0}^Na_n\cos nt $$ with $\pm 1$ coefficients tends to infinity with the degree $N$.
