多圆盘上的Toral代数集与函数论
Toral Algebraic Sets and Function Theory on Polydisks
摘要：
环代数集$A$是$\C^n$中的一个代数集，它与$\T^n$的交集足够大，可以确定$A$上的全纯函数。我们发展了这些集的理论，并给出了一些应用的函数理论在几个变量和运营商理论模型论。特别是，我们表明，唯一性集的极值Pick问题的bidisk是一个toral代数集，合理的内部功能有零集的不可约组件是不是toral，和模型论的可交换对收缩有限缺陷自然生活在一个toral代数集。
英文摘要：
A toral algebraic set $A$ is an algebraic set in $\C^n$ whose intersection with $\T^n$ is sufficiently large to determine the holomorphic functions on $A$. We develop the theory of these sets, and give a number of applications to function theory in several variables and operator theoretic model theory. In particular, we show that the uniqueness set for an extremal Pick problem on the bidisk is a toral algebraic set, that rational inner functions have zero sets whose irreducible components are not toral, and that the model theory for a commuting pair of contractions with finite defect lives naturally on a toral algebraic set.
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