We formulate and prove an analog of Poonen's finite-field Bertini theorem with Taylor conditions that holds in the Grothendieck ring of varieties. This gives a broad generalization of the work of Vakil-Wood, who treated the case of smooth hypersurface sections. In fact, our techniques give analogs in motivic statistics of all known results in arithmetic statistics that have been proven using Poonen's sieve, including work of Bucur-Kedlaya on complete intersections and Erman-Wood on semi-ample Bertini theorems. A key ingredient is the use of motivic Euler products, as introduced by the first author, to write down candidate motivic probabilities. We also formulate a conjecture on the uniform convergence of zeta functions that unifies motivic and arithmetic statistics for varieties over finite fields.

中文翻译：

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动机统计中的动机欧拉积

我们制定并证明了具有泰勒条件的 Poonen 有限域 Bertini 定理的模拟，该定理在格洛腾迪克环中成立。这对 Vakil-Wood 的工作进行了广泛的概括，他处理了光滑超曲面部分的情况。事实上，我们的技术在算术统计中的所有已知结果的动机统计中提供了类似物，这些结果已使用 Poonen 筛子证明，包括 Bucur-Kedlaya 在完全交叉点上的工作和 Erman-Wood 在半充足 Bertini 定理上的工作。一个关键因素是使用第一作者介绍的动机欧拉积来写下候选动机概率。我们还提出了一个关于 zeta 函数一致收敛的猜想，该猜想统一了有限域上变体的动机统计和算术统计。