论文标题
有估值的田地，具有总残留图
Valued fields with a total residue map
论文作者
论文摘要
当$ k $是有限的字段时，贝克尔 - 戴内夫·莱普斯茨（Becker-Denef-Lipschitz（1979））观察到，总残留物映射$ \ text {res}：k（\！（t）\！）\ to k $，它可以在带有$ t $ $ t $的参数的戒指的语言中挑选出laurent系列的常数术语。在这个观察结果的驱动下，我们研究了$ \ text {vf} _ {\ text {res}，i} $的估算字段的$，配备了线性表单$ \ text {res}：k \ t tok $ t to k $，to k $，专门针对评估环上的残基映射。我们证明$ \ text {vf} _ {\ text {res}，i} $不接受模型伴侣。此外，我们证明了Power系列字段$（k（\！（t）\！），\ text {res}）$，配备了如此总的残留图，每当$ k $是无限字段时，都无法确定。结果，我们得到$（\ mathbb {c}（\！（t）\！），\ text {res} _0）$是不可证实的，其中$ \ text {res} _0：\ mathbb {c}（c} \ text {res} _0（f）$ maps $ f $以$ 0 $ $ 0。
When $k$ is a finite field, Becker-Denef-Lipschitz (1979) observed that the total residue map $\text{res}:k(\!(t)\!)\to k$, which picks out the constant term of the Laurent series, is definable in the language of rings with a parameter for $t$. Driven by this observation, we study the theory $\text{VF}_{\text{res},ι}$ of valued fields equipped with a linear form $\text{res}:K\to k$ which specializes to the residue map on the valuation ring. We prove that $\text{VF}_{\text{res},ι}$ does not admit a model companion. In addition, we show that the power series field $(k(\!(t)\!),\text{res})$, equipped with such a total residue map, is undecidable whenever $k$ is an infinite field. As a consequence, we get that $(\mathbb{C}(\!(t)\!), \text{Res}_0)$ is undecidable, where $\text{Res}_0:\mathbb{C}(\!(t)\!)\to \mathbb{C}:f\mapsto \text{Res}_0(f)$ maps $f$ to its complex residue at $0$.
