We improve results of Belair, Macintyre, and Scanlon on valued fields with a valuation preserving automorphism by weakening their assumptions on the residue difference field. In the equicharacteristic zero case we also determine the induced structure on the residue difference field and the value group.
Construction of a valued field whose valuation ring is definable but neither $\exists\forall\exists$ nor $ \forall\exists\forall$-definable in the language of rings
We give a short argument why the tensor product valuation on $K \otimes_k L$ is multiplicative when $k$ is an algebraically closed valued field and $K$ and $L$ are valued extensions (all valuations being in $\bR$). When the valuation on $k$ is non trivial we use the fact that $ACVF$, the theory of algebraically closed (non trivially) valued fields, has quantifier elimination.
Let $ k $ be a field, $ G $ a totally ordered abelian group and $ \mathbb K = k((G)) $ the maximal field of generalised power series, endowed with the canonical valuation $ v $. We study the group $ v \mathrm{-Aut} K $ of valuation preserving automorphisms of a subfield $ k(G)\subseteq K\subseteq \mathbb K $, where $ k(G) $ is the fraction field of the group ring $ k[G] $. Under the assumption that $ K $ satisfies two lifting properties we are able to generalise and refine Hofberger's decomposition of $ v \mathrm{-Aut}\mathbb K $ and prove a structure theorem decomposing $ v\mathrm{-Aut} K $ i…
We prove quantifier elimination for the theory of quasi-real closed fields with a compatible valuation. This unifies the same known results for algebraically closed valued fields and real closed valued fields.
Valuations in Gödel Logic, and the Euler Characteristic
使用V. Klee和G.- C. Rota在六十年代，我们定义了哥德尔逻辑中公式的欧拉特征（在无穷多个真值上）。然后，我们证明了编码的信息的欧拉特征是经典的，即符合类似的概念定义的布尔逻辑。在此基础上，我们定义了公式$φ$的欧拉特征线的多值版本，并证明了它们确实提供了关于$φ$在哥德尔逻辑中的逻辑状态的信息。具体地说，我们的第一个主要结果表明，多值欧拉特征是分离的多值重言式从非重言式的不变量。此外，我们提供了这些广义特征的线性结构的初步调查。我们的第二个主要结果是多值特征的集合形成了一个线性独立集在实向量空间中的所有赋值的哥德尔逻辑在许多命题变量。
In this article, we study the structure of finitely ramified mixed characteristic valued fields. For any two complete discrete valued fields $K_1$ and $K_2$ of mixed characteristic with perfect residue fields, we show that if the $n$-th residue rings are isomorphic for each $n\ge 1$, then $K_1$ and $K_2$ are isometric and isomorphic. More generally, for $n_1\ge 1$, there is $n_2$ depending only on the ramification indices of $K_1$ and $K_2$ such that any homomorphism from the $n_1$-th residue ring of $K_1$ to the $n_2$-th residue ring of $K_2$ can be lifted to a homomorphism between the valuat…
These notes form part of a joint research project on the logic of fields with many valuations, connected by a product formula. We define such structures and name them {\em globally valued fields} (GVFs). This text aims primarily at a proof that {\em the canonical GVF structure on $k(t)^{alg}$ is existentially closed}. This can be read as saying that a variety {\em with a distinguished curve class} is a good approximation for a formula in the language of GVFs, in the same way that a variety is close to a formula for the theory ACF of algebraically closed fields.
