不可数范畴伪有限结构中的计数
Counting in Uncountably Categorical Pseudofinite Structures
AI总结：
本文证明不可数范畴伪有限结构中可定义子集的伪有限基数可表示为强极小集大小的有理多项式，并据此将满足同一不可数范畴理论的有限结构类刻画为多项式R-mec和N维渐近类。
摘要：
我们证明，不可数范畴伪有限结构的每个可定义子集都具有伪有限基数，该基数是任意强极小集大小的有理系数多项式，且多项式的次数等于该子集的莫利秩。基于这一事实，我们证明：若一类有限结构的超积均满足同一不可数范畴理论，则这些有限结构类既是多项式R-mec，也是N维渐近类，其中N为该理论的莫利秩。
英文摘要：
We show that every definable subset of an uncountably categorical pseudofinite structure has pseudofinite cardinality which is polynomial (over the rationals) in the size of any strongly minimal subset, with the degree of the polynomial equal to the Morley rank of the subset. From this fact, we show that classes of finite structures whose ultraproducts all satisfy the same uncountably categorical theory are polynomial $R$-mecs as well as $N$-dimensional asymptotic classes, where $N$ is the Morley rank of the theory.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
