The maximum modulus set of a polynomial

TL;DR

This paper explores the structure of the maximum modulus set of polynomials, constructing examples with prescribed finite features and proving limitations on infinite discontinuities.

Contribution

It extends previous results by constructing polynomials with multiple prescribed discontinuities and singleton components in their maximum modulus sets.

Findings

Constructed polynomials with specified finite discontinuities

Constructed polynomials with singleton components at given points

Proved impossibility of infinitely many discontinuities in the maximum modulus set

Abstract

We study the maximum modulus set, $\mathcal{M}(p)$, of a polynomial $p$. We are interested in constructing $p$ so that $\mathcal{M}(p)$ has certain exceptional features. Jassim and London gave a cubic polynomial $p$ such that $\mathcal{M}(p)$ has one discontinuity, and Tyler found a quintic polynomial $\tilde{p}$ such that $\mathcal{M}(\tilde{p})$ has one singleton component. These are the only results of this type, and we strengthen them considerably. In particular, given a finite sequence $a_1, a_2, \ldots, a_n$ of distinct positive real numbers, we construct polynomials $p$ and $\tilde{p}$ such that $\mathcal{M}(p)$ has discontinuities of modulus $a_1, a_2, \ldots, a_n$, and $\mathcal{M}(\tilde{p})$ has singleton components at the points $a_1, a_2, \ldots, a_n$. Finally we show that these results are strong, in the sense that it is not possible for a polynomial to have infinitely many discontinuities in its maximum modulus set.