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On the zeros of reciprocal Littlewood polynomials

Volume 219 / 2025

Benjamin Bedert Acta Arithmetica 219 (2025), 297-330 MSC: Primary 26C10; Secondary 11C08, 30C15, 41A17 DOI: 10.4064/aa231207-20-4 Published online: 23 June 2025

Abstract

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 $N$.

Authors

  • Benjamin BedertMathematical Institute
    University of Oxford
    Oxford, OX2 6GG, UK
    e-mail

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