论文标题
封闭有序的差分领域中尺寸的巧合
Coincidence of dimensions in closed ordered differential fields
论文作者
论文摘要
从M. Singer的意义上讲，令$ \ Mathcal k = \ langle \ Mathcal r，Δ\ rangle $是一个封闭的有序差分字段，其常数为$ c $。在本说明中，我们证明，对于对$ \ mathcal m = \ langle \ mathcal r，c \ rangle $，$Δ$ dimension和大尺寸重合的集合。作为一个应用程序，我们表征了$ \ Mathcal k $中的可定义集，这些集合在$ c $的内部为$ \ Mathcal M $且具有$δ$ -Dimension $ 0 $。我们进一步表明，对于在$ \ Mathcal K $中可定义的集合，具有$δ$ -Dimension $ 0 $通常并不意味着在$ c $中的共同分析性（与TransSeries相反）。我们还指出，尺寸的巧合也存在于差异封闭的字段和跨性别的背景下。
Let $\mathcal K=\langle\mathcal R, δ\rangle$ be a closed ordered differential field, in the sense of M. Singer, and $C$ its field of constants. In this note, we prove that, for sets definable in the pair $\mathcal M=\langle \mathcal R, C\rangle$, the $δ$-dimension and the large dimension coincide. As an application, we characterize the definable sets in $\mathcal K$ that are internal to $C$ as those sets that are definable in $\mathcal M$ and have $δ$-dimension $0$. We further show that, for sets definable in $\mathcal K$, having $δ$-dimension $0$ does not generally imply co-analyzability in $C$ (in contrast to the case of transseries). We also point out that the coincidence of dimensions also holds in the context of differentially closed fields and in the context of transseries.
