论文标题
计数类别中随机对象的功能
Counting Functions for Random Objects in a Category
论文作者
论文摘要
在算术统计和分析数理论中，计数函数的渐近增长速率为$ x $以下的对象数量数量为$ x \ to \ infty $。我们定义了一般计数函数，这些功能在某些订购下从类别上的对象中计算出表达。鉴于概率度量$μ$ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ x $倾向于$ x $倾向于以$ \ nifty $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ x $ $ x $倾向于$ x $的渐近增长率，因此我们证明了大量的定律版本。这种计数功能是由算术统计数据中的工作激励的，包括数字字段，如Malle的猜想和点数，如BatyRev-Manin猜想所示。锯蛋白的最新工作 - 木材提供了足够的条件，可以在非常广泛的环境中从行为良好的有限矩构建这样的量度$μ$，我们在这种广泛的背景下证明了我们的结果，并额外的假设是尊重该类别中的产品结构。这些结果使我们能够对一般环境中计数功能进行庞大的启发式预测形式化。
In arithmetic statistics and analytic number theory, the asymptotic growth rate of counting functions giving the number of objects with order below $X$ is studied as $X\to \infty$. We define general counting functions which count epimorphisms out of an object on a category under some ordering. Given a probability measure $μ$ on the isomorphism classes of the category with sufficient respect for a product structure, we prove a version of the Law of Large Numbers to give the asymptotic growth rate as $X$ tends towards $\infty$ of such functions with probability $1$ in terms of the finite moments of $μ$ and the ordering. Such counting functions are motivated by work in arithmetic statistics, including number field counting as in Malle's conjecture and point counting as in the Batyrev-Manin conjecture. Recent work of Sawin--Wood gives sufficient conditions to construct such a measure $μ$ from a well-behaved sequence of finite moments in very broad contexts, and we prove our results in this broad context with the added assumption that a product structure in the category is respected. These results allow us to formalize vast heuristic predictions about counting functions in general settings.
