发表机构
Changwon National University（昌原国立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究有限分歧亨泽尔值域域的结构，通过高长度剩余环证明近似提升定理并确定最优误差界，进而得到若干 Ax-Kochen-Ershov 原理及公式等价性结果。
AI 中文摘要
我们研究了混合特征、具有任意剩余域且有限分歧的亨泽尔值域域的结构，通过其高长度剩余环进行，其中长度为 $n$ 的剩余环是赋值环对其极大理想 $n$ 次幂的商环。我们证明了高长度剩余环之间同态的近似提升定理，并明确确定了提升误差的最优界。作为应用，我们利用剩余环上的纯环结构，获得了关于相对完备性、相对存在完备性和存在封闭性的若干 Ax-Kochen-Ershov 原理。我们还证明了，对于所有混合特征 $(0,p)$ 且具有相同初始分歧指数 $e$ 的有限分歧亨泽尔值域域，剩余环上长度为 $k$ 的任何公式等价于纯有序群结构中的句子正规形和 $n$ 层特殊公式，且这种等价性一致成立。这里，对于 $n\ge k$，$n$ 层特殊公式表示长度为 $k$ 的剩余环上的可定义集，该集合由长度为 $n$ 的剩余环上纯环结构中可定义集的投影像给出。此外，这样的 $n$ 是根据提升误差的精确估计最优计算得到的，且仅依赖于 $k$、剩余特征 $p$ 和初始分歧指数 $e$。
英文摘要
We study the structure of finitely ramified henselian valued fields of mixed characteristic with arbitrary residue fields via their residue rings of higher length, where a residue ring of length $n$ is the quotient ring of the valuation ring by the $n$th power of its maximal ideal. We prove an approximate lifting theorem for homomorphisms between residue rings of higher length and explicitly determine the optimal bound of error of lifting. As applications, we obtain several Ax-Kochen-Ershov principles for relative completeness, relative existential completeness, and existential closedness using the pure ring structures on the residue rings. And we show that any formula on a residue ring of length $k$ is equivalent to a normal form of sentences in pure ordered group structure and special formulas at level $n$, uniformly for all finitely ramified henselian valued fields of mixed characteristic $(0,p)$ and the same initial ramification index $e$. Here, for $n\ge k$, a special formula at level $n$ represents a definable set on a residue ring of length $k$ given by a projection image of a definable set in pure ring structure on a residue ring of length $n$. Also, such $n$ is optimally computed from the precise estimation of error of lifting and depends only on $k$, residue characteristic $p$, and initial ramification index $e$.
Comments27 pages
