The maximum modulus set of a polynomial

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.