Polynomials with Maximum Lead Coefficient Bounded on a Finite Set

TL;DR

This paper investigates the maximum lead coefficient of degree d polynomials constrained on a finite set, providing explicit calculations for low degrees and an algorithm for general cases.

Contribution

It introduces a unique polynomial with maximum lead coefficient under bounded conditions and offers explicit formulas and an algorithm for its construction.

Findings

Explicit lead coefficient formulas for degrees up to 4

Algorithm for generating the polynomial for any degree and set

Characterization of the polynomial as unique maximum lead coefficient solution

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 $[x_k]$.