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

Peter Borwein, Tamás Erdélyi, and Géza Kós. "The multiplicity of the zero at 1 of polynomials with constrained coefficients." Acta Arithmetica 159.4 (2013): 387-395. <http://eudml.org/doc/279367>.

@article{PeterBorwein2013,
abstract = {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\} $1/p$, $aj ∈ ℂ$, $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.},
author = {Peter Borwein, Tamás Erdélyi, Géza Kós},
journal = {Acta Arithmetica},
keywords = {polynomials with constrained coefficients; heights of polynomials; zeros of polynomials; bounds for the multiplicity; density of square free numbers},
language = {eng},
number = {4},
pages = {387-395},
title = {The multiplicity of the zero at 1 of polynomials with constrained coefficients},
url = {http://eudml.org/doc/279367},
volume = {159},
year = {2013},
}

TY - JOUR
AU - Peter Borwein
AU - Tamás Erdélyi
AU - Géza Kós
TI - The multiplicity of the zero at 1 of polynomials with constrained coefficients
JO - Acta Arithmetica
PY - 2013
VL - 159
IS - 4
SP - 387
EP - 395
AB - 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} $1/p$, $aj ∈ ℂ$, $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.
LA - eng
KW - polynomials with constrained coefficients; heights of polynomials; zeros of polynomials; bounds for the multiplicity; density of square free numbers
UR - http://eudml.org/doc/279367
ER -