论文标题
功能领域中复合位置的密度以及对真实霍分环应用的应用
Density of Composite Places in Function Fields and Applications to Real Holomorphy Rings
论文作者
论文摘要
鉴于代数功能字段$ f | k $和$ k $上的一个place $ \ wp $，我们证明，在$ f $的所有地方，$ k $ lie lie lie like lie lie lie lie like like like like suftensions in $ k $ a $ k $ a $ wp $。我们将结果应用于$ k = r $的任何实际闭合场，而$ r $的固定位置是其自然的（最好的）真实位置。这导致了对$ f $的真正荷兰戒指的新描述，这可以看作是对Artin对希尔伯特第17个问题的某种改进的类似描述。我们还确定了所有$ f $的拓扑空间$ m（f）$的关系（$ f $的所有$ \ r $ - $ \ r $中包含的剩余场的位置），其所有$ \ r $ f $的子空间与自然$ \ r $ $ r $ $ r $的$ f $的子空间，以及$ r $ $ r $的位置，以及所有$ r $ r $ r $ r $ r $ r $ r的拓扑空间。证明了有关这些空间以及各种相对实际的霍明型环的进一步结果。在本文的结论中，环的真实光谱理论将用于从该角度解释基本概念，并表明空间$ m（f）$只有有限的许多拓扑成分。
Given an algebraic function field $F|K$ and a place $\wp$ on $K$, we prove that the places that are composite with extensions of $\wp$ to finite extensions of $K$ lie dense in the space of all places of $F$, in a strong sense. We apply the result to the case of $K=R$ any real closed field and the fixed place on $R$ being its natural (finest) real place. This leads to a new description of the real holomorphy ring of $F$ which can be seen as an analogue to a certain refinement of Artin's solution of Hilbert's 17th problem. We also determine the relation between the topological space $M(F)$ of all $\R$-places of $F$ (places with residue field contained in $\R$), its subspace of all $\R$-places of $F$ that are composite with the natural $\R$-place of $R$, and the topological space of all $R$-rational places. Further results about these spaces as well as various classes of relative real holomorphy rings are proven. At the conclusion of the paper the theory of real spectra of rings will be applied to interpret basic concepts from that angle and to show that the space $M(F)$ has only finitely many topological components.
