The multiplicity of the zero at 1 of polynomials with constrained coefficients
Volume 159 / 2013
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.
