论文标题
通过概率镜头观察的谐波多项式的价
The valence of harmonic polynomials viewed through the probabilistic lens
论文作者
论文摘要
我们证明，复杂多项式的存在$ p（z）$ $ n $和$ q（z）$ $ m <n $的$ q（z）$，使得和谐波polyenmial $ p（z） + \ overline {q（z）} $至少具有至少$ \ lceil n \ lceil n \ sqrt {m} {m} {m} \ rceil $ rceil $多Zeros。这为Wilmshurst的猜想提供了一系列新的反例，即谐波多项式的最大价值$ P（z）+\ edline {q（z）} $对多项式$ n $ $ n $ $ n $和$ q $ $ n $ n $和$ q $ $ m $ $ m $ m $ m $ is $ m $ is $ m $ is $ m（m-1）+3nn-2 $。更广泛地说，这些示例表明，没有统一（以$ m $）的增长率在价上的线性（以$ n $为单位）。该结果的证明使用了基于估计某个随机谐波多项式系列的平均零数的概率技术。
We prove the existence of complex polynomials $p(z)$ of degree $n$ and $q(z)$ of degree $m<n$ such that the harmonic polynomial $ p(z) + \overline{q(z)}$ has at least $\lceil n \sqrt{m} \rceil$ many zeros. This provides an array of new counterexamples to Wilmshurst's conjecture that the maximum valence of harmonic polynomials $p(z)+\overline{q(z)}$ taken over polynomials $p$ of degree $n$ and $q$ of degree $m$ is $m(m-1)+3n-2$. More broadly, these examples show that there does not exist a linear (in $n$) bound on the valence with a uniform (in $m$) growth rate. The proof of this result uses a probabilistic technique based on estimating the average number of zeros of a certain family of random harmonic polynomials.
