发表机构
Boğaziçi Üniversitesi; Instytut Matematyczny Uniwersytet Wrocławski（博阿齐奇大学; 弗罗茨瓦夫大学数学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究域论添加自同构后模型伴生的存在性，证明两类情形（有界PAC域及ACFA的完备化）添加自同构后均无模型伴生，并给出显式障碍。
AI 中文摘要
我们在两个互补的背景下研究了在域论中添加自同构后模型伴生的存在性。对于特征为零的有界PAC域，若其包含所有单位根并允许一个素数阶的循环伽罗瓦商，则适当选择可数个常数可得到一个模型完全的域论，但在添加固定这些命名常数的自同构后，该域论没有模型伴生。特别地，当绝对伽罗瓦群为自由pro有限群或有限正秩的自由pro-$p$群时，此结论适用。其次，对于$\text{ACFA}$的每一个完备化（在任意特征下），添加第二个交换自同构后没有模型伴生。这两个显式障碍使用了一个沿无限轨道的惯性自同构的公共有序乘积，分别结合了一个等变PAC完备化和一个代数闭差分域嵌入。
英文摘要
We study the existence of model companions after adjoining an automorphism to field theories in two complementary settings. For bounded PAC fields of characteristic zero containing all roots of unity and admitting a cyclic Galois quotient of prime order, a suitable choice of countably many constants gives a model-complete field theory which has no model companion after adding an automorphism fixing the named constants. This applies, in particular, when the absolute Galois group is free profinite or free pro-$p$ of finite positive rank. Secondly, for every completion of $\ACFA$ in every characteristic, adjoining a second commuting automorphism has no model companion. The two explicit obstructions use a common ordered product of inertia automorphisms along an infinite orbit, combined respectively with an equivariant PAC completion and an algebraically closed difference-field embedding.
