Model theory of valued fields with endomorphism

提出了非满射自同态的值域的模型理论研究。具有自同构的值域的模型理论，特别是由bsamlair， Macintyre和Scanlon处理的Witt Frobenius案例，在过去的15年中得到了广泛的发展，例如，得到了各种类型的 $\sigma$-henselian值差域。我们计划将这些结果推广到非满射环境。非满射自同态最自然的例子是不完全域上的Frobenius映射。与Witt Frobenius类似，我们研究的主要例子是赋予Frobenius升力的不完全残馀场上的Cohen场。科恩场的模型理论是最近在安斯科姆和扬克的工作中发展起来的。为了在我们的设置中获得Ax-Kochen-Ershov型结果，有必要首先了解等特征0情况。这个例子本身就很有趣，因为它包含了具有Frobenius lift的Cohen场的渐近理论。我们特别感兴趣的是获得模型理论驯服概念从值群和剩余域到值差域的相对完备性和转移，以及识别各种子类的模型同伴。另一个自然的例子 $\sigma$-henselian值差分场由可分闭值场与Frobenius的超积给出。在这里，自同构不再是等距，而值群的诱导自同构是 $\omega$-增加。通过Chatzidakis和Hrushovski的工作，将这类超积的残差场作为一个有区别的自同态场而存在封闭。我们的目的是证明值差域的类似结果，即它在自然语言中是存在封闭的，并由此推断出存在性和模型同伴的公理化。

We propose a model-theoretic investigation of valued fields with non-surjective endomorphism. The model theory of valued fields with automorphism, in particular the Witt Frobenius case treated by Bélair, Macintyre and Scanlon, was extensively developed over the last 15 years, e.g., obtaining Ax-Kochen-Ershov principles for various classes of $\sigma$-henselian valued difference fields.We plan to generalize these results to the non-surjective context. The most natural example of a non-surjective endomorphism is the Frobenius map on an imperfect field. In analogy to the Witt Frobenius, the main examples for our study are Cohen fields over imperfect residue fields endowed with a lift of the Frobenius. The model theory of Cohen fields was recently developed in the work of Anscombe and Jahnke. In order to obtain Ax-Kochen-Ershov type results in our setting, it will be necessary to first understand the equicharacteristic 0 case. This case is interesting in its own right, as it encompasses the asymptotic theory of Cohen fields with Frobenius lift. We are particularly interested in obtaining relative completeness and transfer of model theoretic tameness notions from value group and residue field to the valued difference field, as well as in identifying model companions for various subclasses.Another natural example of a $\sigma$-henselian valued difference fields is given by ultraproducts of separably closed valued fields with Frobenius. Here, the endomorphism is no longer an isometry, and the induced automorphism of the value group is $\omega$-increasing. By work of Chatzidakis and Hrushovski the residue difference field of such an ultraproduct is existentially closed as a field with distinguished endomorphism. We aim to show the analogous result for the valued difference field, namely that it is existentially closed in a natural language, and infer existence and an axiomatization of the model-companion from this.