In this article we study definable functions in tame expansions of algebraically closed valued fields. For a given definable function we have two types of results: type (I), which hold in a neighborhood of infinity, and type (II), which hold locally for all but finitely many points in the domain of the function. In the first part of the article, we show type (I) and (II) results concerning factorizations of definable functions over the value group. As an application, we show that tame expansions of algebraically closed valued fields having value group ℚ (like ℂ p and $$\overline{\mathbb{F}_p}^{\rm{alg}}((t\mathbb{C}))$$ ) are polynomially bounded. In the second part, under an additional assumption on the asymptotic behavior of unary definable functions of the value group, we lift these factorizations over the residue multiplicative structure RV. In characteristic 0, we obtain as a corollary that the domain of a definable function f : X ⊆ K → K can be partitioned into sets F ∪ E ∪ J , where F is finite, f E is locally constant and f J satisfies locally the Jacobian property.

中文翻译：

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代数闭值域的驯服扩展中的可定义函数

在本文中，我们研究代数闭值域的驯服展开中的可定义函数。对于给定的可定义函数，我们有两种类型的结果：类型 (I)，它在无穷大的邻域中成立，以及类型 (II)，它在函数域中除有限点之外的所有点都在本地成立。在文章的第一部分，我们展示了关于值组上可定义函数的因式分解的类型 (I) 和 (II) 结果。作为一个应用，我们展示了具有值组 ℚ 的代数闭值域的驯服扩展（如 ℂ p 和 $$\overline{\mathbb{F}_p}^{\rm{alg}}((t\mathbb{C }))$$ ) 是多项式有界的。在第二部分中，在对值组的一元可定义函数的渐近行为的附加假设下，我们将这些因式分解提升到残差乘法结构 RV 上。