论文标题
对有序字段的可确定估值
Definable valuations on ordered fields
论文作者
论文摘要
我们研究了对有序领域的凸价估值的确定性，特别关注了Henselian估值的杰出子类。在有序字段的设置中，人们可以在戒指的语言中考虑确定性$ \ mathcal {l} _ {\ mathrm {r}} $，以及有序戒指的较丰富语言$ \ mathcal {l} _ {\ mathrm {\ sathrm {or}} $。我们分析和比较两种语言的可确定性，并显示以下相反的结果：虽然有凸的估值可以在语言$ \ mathcal {l} _ {\ mathrm {or}} $中定义，但不在语言$ \ nathcal $ \ mathcal {l} _ {l} _ {\ mathrm {\ mathrm {\ mathrm {\ mathrm { $ \ MATHCAL {l} _ {\ MATHRM {或}} $ - 可定义的Henselian估值已经为$ \ Mathcal {l} _ {\ Mathrm {r}} $ - definable。为了证明后者，我们证明了有序的henselian估值场的值组和有序的残基字段被稳定地嵌入（分别作为有序的Abelian组，作为有序场）。此外，我们表明，在几乎真实的封闭字段中，任何$ \ mathcal {l} _ {\ mathrm {或}} $ - 可定义的估值是Henselian。
We study the definability of convex valuations on ordered fields, with a particular focus on the distinguished subclass of henselian valuations. In the setting of ordered fields, one can consider definability both in the language of rings $\mathcal{L}_{\mathrm{r}}$ and in the richer language of ordered rings $\mathcal{L}_{\mathrm{or}}$. We analyse and compare definability in both languages and show the following contrary results: while there are convex valuations that are definable in the language $\mathcal{L}_{\mathrm{or}}$ but not in the language $\mathcal{L}_{\mathrm{r}}$, any $\mathcal{L}_{\mathrm{or}}$-definable henselian valuation is already $\mathcal{L}_{\mathrm{r}}$-definable. To prove the latter, we show that the value group and the ordered residue field of an ordered henselian valued field are stably embedded (as an ordered abelian group, respectively as an ordered field). Moreover, we show that in almost real closed fields any $\mathcal{L}_{\mathrm{or}}$-definable valuation is henselian.
