Coppersmith-Rivlin type inequalities and the order of vanishing of polynomials at 1

"Coppersmith-Rivlin type inequalities and the order of vanishing of polynomials at 1." Acta Arithmetica 172.3 (2016): 271-284. <http://eudml.org/doc/279368>.

@article{Unknown2016,
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 $P(x) = ∑_\{j=0\}^n\{a_jx^j\}$, $|a_0| ≥ L(∑_\{j=1\}^n |a_j|^p)^\{1/p\}$, $a_j ∈ ℂ$, such that $(x-1)^k$ divides P(x). For n ∈ ℕ, L > 0, and q ≥ 1 let $μ_q(n,L)$ be the smallest value of k for which there is a polynomial Q of degree k with complex coefficients such that $|Q(0)| > 1/L (∑_\{j=1\}^n |Q(j)|^q)^\{1/q\}$. We find the size of $κ_p(n,L)$ and $μ_q(n,L)$ for all n ∈ ℕ, L > 0, and 1 ≤ p,q ≤ ∞. The result about $μ_∞(n,L)$ is due to Coppersmith and Rivlin, but our proof is completely different and much shorter even in that special case.},
journal = {Acta Arithmetica},
keywords = {polynomials; restricted coefficients; order of vanishing at 1; Coppersmith-Rivlin type inequalities},
language = {eng},
number = {3},
pages = {271-284},
title = {Coppersmith-Rivlin type inequalities and the order of vanishing of polynomials at 1},
url = {http://eudml.org/doc/279368},
volume = {172},
year = {2016},
}

TY - JOUR
TI - Coppersmith-Rivlin type inequalities and the order of vanishing of polynomials at 1
JO - Acta Arithmetica
PY - 2016
VL - 172
IS - 3
SP - 271
EP - 284
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 $P(x) = ∑_{j=0}^n{a_jx^j}$, $|a_0| ≥ L(∑_{j=1}^n |a_j|^p)^{1/p}$, $a_j ∈ ℂ$, such that $(x-1)^k$ divides P(x). For n ∈ ℕ, L > 0, and q ≥ 1 let $μ_q(n,L)$ be the smallest value of k for which there is a polynomial Q of degree k with complex coefficients such that $|Q(0)| > 1/L (∑_{j=1}^n |Q(j)|^q)^{1/q}$. We find the size of $κ_p(n,L)$ and $μ_q(n,L)$ for all n ∈ ℕ, L > 0, and 1 ≤ p,q ≤ ∞. The result about $μ_∞(n,L)$ is due to Coppersmith and Rivlin, but our proof is completely different and much shorter even in that special case.
LA - eng
KW - polynomials; restricted coefficients; order of vanishing at 1; Coppersmith-Rivlin type inequalities
UR - http://eudml.org/doc/279368
ER -