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

Peter Borwein ; Tamás Erdélyi ; Géza Kós

Acta Arithmetica, Tome 161 (2013), p. 387-395 / Harvested from The Polish Digital Mathematics Library

Résumé

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.

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Publié le : 2013-01-01

Zbl 1284.11054

EUDML-ID : urn:eudml:doc:279367