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

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}, \