发表机构
Sun Yat-Sen University; Aristotle University of Thessaloniki（中山大学; 塞萨洛尼基亚里士多德大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究单位球中稳定多项式边界零点，借助单位球插值理论等，通过峰值集描述其边界零点，在\(n = 2\)时给出刻画，还应用该理论刻画相关循环多项式，发展了更一般几何设置理论。
AI 中文摘要
单位球中的插值理论和半代数几何给出了稳定多项式边界零点的明确描述。给定一个在单位球中无零点且在球面上沿至多一维子流形消失的多项式\(p\in \mathbb{C}[z_1,...,z_n]\)，我们根据\(A^\infty(\mathbb{B}_n)\)的峰值集来描述边界零点\(\mathcal{Z}(p)\cap\mathbb{S}_n\)。特别地，在\(n = 2\)的情况下，我们通过证明\(\mathcal{Z}(p)\cap\mathbb{S}_2\)的每个聚点都位于一维实解析子流形的相对内部，且这些子流形构成\(\mathcal{Z}(p)\cap\mathbb{S}_2\)非孤立部分的叶状结构来实现一种刻画。作为所发展理论的应用，我们在狄利克雷型空间\(\mathcal{D}_{n - 1/2}(\mathbb{B}_n)\)中得到了没有弱本质奇点的循环多项式的一种刻画。还发展了关于边界零点更一般几何设置的理论，旨在为进一步扩展提供一个起点。
英文摘要
We study polynomials in several complex variables that are stable (i.e. they have no zeros in the unit ball $\mathbb{B}_n$), but vanish on the unit sphere along submanifolds of dimension at most one. We characterize the zeros of such polynomials on the unit sphere in terms of peak sets for the space $A^\infty(\mathbb{B}_n).$ Furthermore, we explicitly construct a polynomial in $\mathbb{C}^3$ that provides a negative answer to the question of whether the equivalent conditions of this characterization universally hold for all stable polynomials. As an application of the developed theory, we obtain a characterization of a certain class of cyclic polynomials in the Dirichlet-type space $D_{n-\frac{1}{2}}(\mathbb{B}_n).$ Next, we examine polynomials whose local zero set, in a neighborhood of points on the unit sphere, coincides with the graph of a holomorphic function. In this particular case, we achieve a characterization of the set of zeros lying on the unit sphere whenever the zero set has local maximum dimension $n-1$. Lastly, we discuss potential generalizations and open problems.
Comments23 pages, 2 figures, v2 current status: corrected errors; improved and added several parts; the analysis on the boundary zeros for n=2 of v1 has been deferred to a subsequent work
