Polynomials with Maximum Lead Coefficient Bounded on a Finite Set

Abstract

What is the maximum possible value of the lead coefficient of a degree $d$ polynomial $Q(x)$ if $|Q(1)|,|Q(2)|,\ldots,|Q(k)|$ are all less than or equal to one? More generally we write $L_{d,[x_k]}(x)$ for what we prove to be the unique degree $d$ polynomial with maximum lead coefficient when bounded between $1$ and $-1$ for $x\in [x_k]=\{x_1,\cdots,x_k\}$. We calculate explicitly the lead coefficient of $L_{d,[x_k]}(x)$ when $d\leq 4$ and the set $[x_k]$ is an arithmetic progression. We give an algorithm to generate $L_{d,[x_k]}(x)$ for all $d$ and $[x_k]$.