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## Acta Arithmetica

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

### Volume 159 / 2013

Acta Arithmetica 159 (2013), 387-395 MSC: 11C08, 41A17, 26C10, 30C15. DOI: 10.4064/aa159-4-7

#### Abstract

For $n \in {\mathbb N}$, $L > 0$, and $p \geq 1$ let $\kappa_p(n,L)$ be the largest possible value of $k$ for which there is a polynomial $P \neq 0$ of the form $$P(x) = \sum_{j=0}^n{a_jx^j}, \quad\ |a_0| \geq L \Big( \sum_{j=1}^n{|a_j|^p} \Big)^{1/p}, \, a_j \in {\mathbb C},$$ such that $(x-1)^k$ divides $P(x)$. For $n \in {\mathbb N}$ and $L > 0$ let $\kappa_\infty(n,L)$ be the largest possible value of $k$ for which there is a polynomial $P \neq 0$ of the form $$P(x) = \sum_{j=0}^n{a_jx^j}, \quad\ |a_0| \geq L \max_{1 \leq j \leq n}{|a_j|}, \, a_j \in {\mathbb C},$$ such that $(x-1)^k$ divides $P(x)$. We prove that there are absolute constants $c_1 > 0$ and $c_2 > 0$ such that $$c_1 \sqrt{n/L} -1 \leq \kappa_{\infty}(n,L) \leq c_2 \sqrt{n/L}$$ for every $L \geq 1$. This complements an earlier result of the authors valid for every $n \in {\mathbb N}$ and $L \in (0,1]$. Essentially sharp results on the size of $\kappa_2(n,L)$ are also proved.

#### Authors

• Peter BorweinDepartment of Mathematics and Statistics
Simon Fraser University
e-mail
• Tamás ErdélyiDepartment of Mathematics
Texas A&M University
College Station, TX 77843, U.S.A.
e-mail
• Géza KósMathematical Institute
Lóránd Eötvös University
Pázmány P. s. 1/c
Budapest, Hungary H-1117
and
Computer and Automation Research Institute
Kende u. 13-17
Budapest, Hungary H-1111
e-mail

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