Let $T$ be an algebraic torus over an algebraically closed field, let $X$ be a smooth closed subvariety of a $T$-toric variety such that $U = X \cap T$ is not empty, and let $\mathscr{L}(X)$ be the arc scheme of $X$. We define a tropicalization map on $\mathscr{L}(X) \setminus \mathscr{L}(X \setminus U)$, the set of arcs of $X$ that do not factor through $X \setminus U$. We show that each fiber of this tropicalization map is a constructible subset of $\mathscr{L}(X)$ and therefore has a motivic volume. We prove that if $U$ has a compactification with simple normal crossing boundary, then the generating function for these motivic volumes is rational, and we express this rational function in terms of certain lattice maps constructed in Hacking, Keel, and Tevelev's theory of geometric tropicalization. We explain how this result, in particular, gives a formula for Denef and Loeser's motivic zeta function of a polynomial. To further understand this formula, we also determine precisely which lattice maps arise in the construction of geometric tropicalization.

中文翻译：

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热带化纤维的动力体积

令 $T$ 是代数闭域上的代数环面，令 $X$ 是 $T$-toric 变体的平滑闭子变体，使得 $U = X \cap T$ 不为空，并令 $\mathscr {L}(X)$ 是 $X$ 的弧线方案。我们在 $\mathscr{L}(X) \setminus \mathscr{L}(X \setminus U)$ 上定义了一个热带化地图，这是 $X$ 的一组不通过 $X \setminus U$ 分解的弧。我们表明该热带化地图的每条纤维都是 $\mathscr{L}(X)$ 的可构造子集，因此具有动机量。我们证明如果 $U$ 有一个简单的法线交叉边界的紧化，那么这些动机体积的生成函数是有理的，我们用 Hacking、Keel 和 Tevelev 的几何理论构建的某些格图来表达这个有理函数热带化。我们解释了这个结果如何，特别是，给出多项式的 Denef 和 Loeser 的动机 zeta 函数的公式. 为了进一步理解这个公式，我们还精确地确定了几何热带化的构建中出现了哪些点阵图。