论文标题
多项式的最大模量集
The maximum modulus set of a polynomial
论文作者
论文摘要
我们研究了多项式$ p $的最大模量集，$ \ mathcal {m}（p）$。我们有兴趣构建$ P $，因此$ \ Mathcal {M}（P）$具有某些非凡功能。贾西姆（Jassim）和伦敦（London）给出了立方多项式$ p $，以使$ \ mathcal {m}（p）$有一种不连续性，而泰勒（Tyler）发现了一个Quintic polyenmial $ \ tilde {p} $，因此$ \ nathcal {m} m}（m}（\ tilde {p}）$有一个singleton component。这些是这种类型的唯一结果，我们可以大大加强它们。特别是，给定有限的序列$ a_1，a_2，\ ldots，a_n $具有独特的实际实数，我们构造多项式$ p $和$ \ tilde {p} $，以至于$ \ m athcal {m}（m} $ $ \ MATHCAL {M}（\ tilde {p}）$在点$ a_1，a_2，\ ldots，a_n $的singleton组件。 最后，我们证明了这些结果很强，从某种意义上说，多项式不可能在其最大模量集中具有无限的许多不连续性。
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.
