Counting Functions for Random Objects in a Category

IF 0.6 4区 数学 Q3 MATHEMATICS
Brandon Alberts
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引用次数: 0

Abstract

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 \(X\rightarrow \infty \). We define general counting functions which count epimorphisms out of an object on a category under some ordering. Given a probability measure \(\mu \) 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 \(\mu \) 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 \(\mu \) 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.

类别中随机对象的计数函数
在算术统计和解析数论中,研究了给出X以下阶数的计数函数的渐近增长率为\(X\rightarrow \infty \)。我们定义了一般计数函数,用于在一定排序下对范畴上的对象的外胚计数。在充分考虑乘积结构的范畴的同构类上,我们给出了一个概率测度\(\mu \),证明了大数定律的一个版本,给出了关于\(\mu \)的有限矩和排序的概率为1的函数在X趋向\(\infty \)时的渐近增长率。这样的计数函数是由算术统计中的工作激发的,包括马尔猜想中的数域计数和Batyrev-Manin猜想中的点计数。Sawin-Wood最近的工作给出了在非常广泛的背景下从一个良好的有限矩序列构造这样一个测度\(\mu \)的充分条件,并且我们在这个广泛的背景下证明了我们的结果,并增加了一个假设,即范畴内的产品结构是受尊重的。这些结果使我们能够形式化一般情况下计数函数的大量启发式预测。
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来源期刊
CiteScore
1.30
自引率
16.70%
发文量
29
审稿时长
>12 weeks
期刊介绍: Applied Categorical Structures focuses on applications of results, techniques and ideas from category theory to mathematics, physics and computer science. These include the study of topological and algebraic categories, representation theory, algebraic geometry, homological and homotopical algebra, derived and triangulated categories, categorification of (geometric) invariants, categorical investigations in mathematical physics, higher category theory and applications, categorical investigations in functional analysis, in continuous order theory and in theoretical computer science. In addition, the journal also follows the development of emerging fields in which the application of categorical methods proves to be relevant. Applied Categorical Structures publishes both carefully refereed research papers and survey papers. It promotes communication and increases the dissemination of new results and ideas among mathematicians and computer scientists who use categorical methods in their research.
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