多项式的虚投影
Imaginary projections of polynomials
摘要：
我们引入多元多项式$f \in \mathbb{C}[\mathbf{z}]$的虚投影作为$f$的簇在其虚部$\mathcal{I}(f) \ = \ \{\text{Im}(\mathbf{z}) \, : \, \mathbf{z} \in \mathcal{V}(f) \}$上的投影。由于多项式$f$是稳定的当且仅当$\mathcal{I}(f) \cap \mathbb{R}_{>0}^n \ = \ \emptyset$，这一概念提供了一个新的几何视图潜在的多项式的稳定性问题。我们表明，连接组件的补充封闭的想象中的投影是凸的，从而打开一个中心连接到理论的变形虫和coamoebas。在此基础上，建立了补多项式的分支的结构性质，如其最大个数的下界，证明了二次多项式虚投影的完全分类，并刻画了任意次多项式的极限方向.
英文摘要：
We introduce the imaginary projection of a multivariate polynomial $f \in \mathbb{C}[\mathbf{z}]$ as the projection of the variety of $f$ onto its imaginary part, $\mathcal{I}(f) \ = \ \{\text{Im}(\mathbf{z}) \, : \, \mathbf{z} \in \mathcal{V}(f) \}$. Since a polynomial $f$ is stable if and only if $\mathcal{I}(f) \cap \mathbb{R}_{>0}^n \ = \ \emptyset$, the notion offers a novel geometric view underlying stability questions of polynomials. We show that the connected components of the complement of the closure of the imaginary projections are convex, thus opening a central connection to the theory of amoebas and coamoebas. Building upon this, the paper establishes structural properties of the components of the complement, such as lower bounds on their maximal number, proves a complete classification of the imaginary projections of quadratic polynomials and characterizes the limit directions for polynomials of arbitrary degree.
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