Artykuły w formacie PDF dostępne są dla subskrybentów, którzy zapłacili za dostęp online, po podpisaniu licencji Licencja użytkownika instytucjonalnego. Czasopisma do 2009 są ogólnodostępne (bezpłatnie).

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

Tom 192 / 2020

Tamás Erdélyi Acta Arithmetica 192 (2020), 189-210 MSC: 11C08, 41A17, 26C10, 30C15. DOI: 10.4064/aa190204-27-5 Opublikowany online: 25 October 2019


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.


  • Tamás ErdélyiDepartment of Mathematics
    Texas A&M University
    College Station, TX 77843, U.S.A.

Przeszukaj wydawnictwa IMPAN

Zbyt krótkie zapytanie. Wpisz co najmniej 4 znaki.

Przepisz kod z obrazka

Odśwież obrazek

Odśwież obrazek