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.

我们研究多项式p的最大模数集\（{{\ mathcal {M}}}（p）\）。我们对构造p感兴趣，因此\（{{\ mathcal {M}}}（p）\）具有某些特殊功能。Jassim和London给出了三次多项式p，使得\（{{\ mathcal {M}}}（p）\）具有一个不连续点，而Tyler找到了一个五次多项式\（{\ tilde {p}} \）使得\ （{{\ mathcal {M}}}（{\ tilde {p}}）\）具有一个单例分量。这些是这种类型的唯一结果，我们将对其进行大力增强。特别是，给定一个不同的正实数的有限序列\（a_1，a_2，\ ldots，a_n \），我们可以构造多项式p和\（{\ tilde {p}} \），这样\（{{\ mathcal {M}}}（p）\）具有模数\ {a_1，a_2，\ ldots，a_n \）和\ （{{\ mathcal {M}}}（{\ tilde {p}}）\）在点\（a_1，a_2，\ ldots，a_n \）处具有单例分量。最后，从多项式在其最大模数集中不可能具有无限多个不连续的意义上说，这些结果很强。