The multiplicity of the zero at 1 of polynomials with constrained coefficients

• Autorzy: Peter Borwein , Tamás Erdélyi , Géza Kós
• Języki publikacji: EN

Abstrakty

EN

For n ∈ ℕ, L > 0, and p ≥ 1 let $κ_p(n,L)$ be the largest possible value of k for which there is a polynomial P ≠ 0 of the form
$P(x) = ∑_{j=0}^n{a_jx^j}$, $|a_0| ≥ L(∑_{j=1}^n{|a_j|^p} }^{1/p}$, $a_j ∈ ℂ$,
such that $(x-1)^k$ divides P(x). For n ∈ ℕ and L > 0 let $κ_∞(n,L)$ be the largest possible value of k for which there is a polynomial P ≠ 0 of the form
$P(x) = ∑_{j=0}^n{a_jx^j}$, $|a_0| ≥ Lmax_{1 ≤ j ≤ n}{|a_j|}$, $a_j ∈ ℂ$,
such that $(x-1)^k$ divides P(x). We prove that there are absolute constants c₁ > 0 and c₂ > 0 such that
$c_1 √(n/L) -1 ≤ κ_{∞}(n,L) ≤ c_2 √(n/L)$
for every L ≥ 1. This complements an earlier result of the authors valid for every n ∈ ℕ and L ∈ (0,1]. Essentially sharp results on the size of κ₂(n,L) are also proved.

Kategorie tematyczne

• 26C10: Polynomials: location of zeros
• 11C08: Polynomials
• 30C15: Zeros of polynomials, rational functions, and other analytic functions (e.g. zeros of functions with bounded Dirichlet integral)
• 41A17: Inequalities in approximation (Bernstein, Jackson, Nikol\cprime ski{\u\i}-type inequalities)

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Acta Arithmetica
• Rocznik: 2013
• Tom: 159
• Numer: 4
• Strony: 387-395

Daty

wydano

2013

Twórcy

autor

Peter Borwein

• Department of Mathematics and Statistics, Simon Fraser University, Burnaby, BC, Canada V5A 1S6

autor

Tamás Erdélyi

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

autor

Géza Kós

• Mathematical Institute, Lóránd Eötvös University, Pázmány P. s. 1/c, Budapest, Hungary H-1117
• Computer and Automation Research Institute, Kende u. 13-17, Budapest, Hungary H-1111

Identyfikatory

DOI

10.4064/aa159-4-7