差分方程的特化与高 Frobenius 幂
Specialization of Difference Equations and High Frobenius Powers
AI总结：
本文研究带局部无限收缩自同构的赋值域，建立其变换赋值结构、模型伴随及与 Frobenius 极限理论的联系，并推出高 Frobenius 幂下可判定性与 motivic 相交理论。
摘要：
我们研究带有自同构 σ 的赋值域，其中 σ 是局部无限收缩的，即对所有 0<α∈Γ 都有 α≪σα。我们证明赋值论中的各种概念，例如 Henselian 包和严格 Henselian 包，都具有有意义的变换类比。我们证明了典范合并结果，并展示变换野性分歧如何由广义向量群上的 torsor 控制。在模型论方面，我们确定了模型伴随：它是可判定的，具有简单的公理化，并享有直至代数有界量词的量词消去。该模型伴随被证明与 Frobenius 作用在代数闭且非平凡赋值域上的极限理论一致。这为差分簇的 motivic 相交理论开辟了道路，而该理论此前仅在特征零下可用。作为第一个推论，带有指定 Frobenius x↦x^q 的代数闭赋值域类是可判定的，并且对 q 一致。
英文摘要：
We study valued fields equipped with an automorphism $σ$ which is locally infinitely contracting in the sense that $α\llσα$ for all $0<α\inΓ$. We show that various notions of valuation theory, such as Henselian and strictly Henselian hulls, admit meaningful transformal analogues. We prove canonical amalgamation results, and exhibit the way that transformal wild ramification is controlled by torsors over generalized vector groups. Model theoretically, we determine the model companion: it is decidable, admits a simple axiomatization, and enjoys elimination of quantifiers up to algebraically bounded quantifiers. The model companion is shown to agree with the limit theory of the Frobenius action on an algebraically closed and nontrivially valued field. This opens the way to a motivic intersection theory for difference varieties that was previously available only in characteristic zero. As a first consequence, the class of algebraically closed valued fields equipped with a distinguished Frobenius $x\mapsto x^{q}$ is decidable, uniformly in $q$.
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