We prove that Hilbert’s Tenth Problem for meromorphic functions and for entire functions in several variables is unsolvable over the language of rings, together with constant symbols for two of the variables and a predicate for a place. This is the first result in the literature, which proves unsolvability of a diophantine problem for the ring of complex analytic functions in a number of variables. The proof rests upon an interplay between analytic geometry, analysis, number theory, and logic.

中文翻译：

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多变量复亚纯函数的希尔伯特第十问题

我们证明了关于亚纯函数和多个变量中的整个函数的希尔伯特第十问题在环语言上是不可解决的，其中两个变量的常量符号和一个位置的谓词。这是文献中的第一个结果，它证明了在多个变量中复杂解析函数环的丢番图问题的不可解性。证明依赖于解析几何、分析、数论和逻辑之间的相互作用。