Coppersmith–Rivlin type inequalities and the order of vanishing of polynomials at 1
Volume 172 / 2016
Abstract
For $n \in {\mathbb N}$, $L \gt 0$, and $p \geq 1$ let $\kappa_p(n,L)$ be the largest possible value of $k$ for which there is a polynomial $P \not \equiv 0$ of the form $$ P(x) = \sum_{j=0}^n{a_jx^j}, \quad |a_0| \geq L \Bigl( \sum_{j=1}^n{|a_j|^p} \Bigr)^{1/p}, \ \quad a_j \in {\mathbb C}, $$ such that $(x-1)^k$ divides $P(x)$. For $n \in {\mathbb N}$, $L \gt 0$, and $q \geq 1$ let $\mu_q(n,L)$ be the smallest value of $k$ for which there is a polynomial $Q$ of degree $k$ with complex coefficients such that $$ |Q(0)| \gt \frac 1L \Bigl( \sum_{j=1}^n{|Q(j)|^q} \Bigr)^{1/q}. $$ We find the size of $\kappa_p(n,L)$ and $\mu_q(n,L)$ for all $n \in {\mathbb N}$, $L \gt 0$, and ${1 \leq p,q \leq \infty}$. The result about $\mu_\infty(n,L)$ is due to Coppersmith and Rivlin, but our proof is completely different and much shorter even in that special case.
