Heights of squares of Littlewood polynomials and infinite series

• Autorzy: Artūras Dubickas
• Języki publikacji: EN

Abstrakty

EN

Let P be a unimodular polynomial of degree d-1. Then the height H(P²) of its square is at least √(d/2) and the product L(P²)H(P²), where L denotes the length of a polynomial, is at least d². We show that for any ε > 0 and any d ≥ d(ε) there exists a polynomial P with ±1 coefficients of degree d-1 such that H(P²) < (2+ε)√(dlogd) and L(P²)H(P²)< (16/3+ε)d²log d. A similar result is obtained for the series with ±1 coefficients. Let $A_m$ be the mth coefficient of the square f(x)² of a unimodular series $f(x) = ∑_{i=0}^{∞} a_i x^i$, where all $a_i ∈ ℂ$ satisfy $|a_i| = 1$. We show that then $lim sup_{m → ∞} |A_m|/√m ≥ 1$ and that there exist some infinite series with ±1 coefficients and an integer m(ε) such that $|A_m| < (2+ε)√(mlogm)$ for each m ≥ m(ε).

Kategorie tematyczne

• 11B83: Special sequences and polynomials
• 11R09: Polynomials (irreducibility, etc.)

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Annales Polonici Mathematici
• Rocznik: 2012
• Tom: 105
• Numer: 2
• Strony: 145-153

Daty

wydano

2012

Twórcy

autor

Artūras Dubickas

• Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, Vilnius LT-03225,, Lithuania
• Vilnius University Institute of Mathematics and Informatics, Vilnius University, Akademijos 4, Vilnius LT-08663, Lithuania

Identyfikatory

DOI

10.4064/ap105-2-3