价值树
The valuative tree
摘要：
我们描述了环C[[x，y]]上所有实值赋值v的集合V，该集合由min{v（x），v（y）}=1规范化。它具有R-树的自然结构，由序关系v小于v'当且仅当对所有f v（f）小于v'（f）。它也可以被度量化，赋予它一个度量树结构。从代数的观点来看，这些结构是通过对C[[x，y]]的Riemann-Zerkki簇取适当的商，以迫使它成为Hausdorff拓扑空间而得到的。V上的树结构也提供了自然超度量中具有不可约曲线球的赋值的识别。我们表明，对偶图的所有序列的爆破补丁在一起，产生一个R-树自然同构的V。总之，这给了许多不同的方法的价值树V。然后，我们描述了一个自然的拉普拉斯算子V。它关联到（特殊）功能V一个复杂的Borel措施。使用这个算子，我们展示了如何测量的价值树可以用来编码自然的整闭理想R和上同调类的本地模拟的voute etoilee在复平面。
英文摘要：
We describe the set V of all real valued valuations v on the ring C[[x,y]] normalized by min{v(x),v(y)}=1. It has a natural structure of an R-tree, induced by the order relation v is less than v' iff v(f) is less than v'(f) for all f. It can also be metrized, endowing it with a metric tree structure. From the algebraic point of view, these structures are obtained by taking a suitable quotient of the Riemann-Zariski variety of C[[x,y]], in order to force it to be a Hausdorff topological space. The tree structure on V also provides an identification of valuations with balls of irreducible curves in a natural ultrametric. We show that the dual graphs of all sequences of blow-ups patch together, yielding an R-tree naturally isomorphic to V. Altogether, this gives many different approaches to the valuative tree V. We then describe a natural Laplace operator on V. It associates to (special) functions of V a complex Borel measure. Using this operator, we show how measures on the valuative tree can be used to encode naturally both integrally closed ideals in R and cohomology classes of the local analog of voute etoilee over the complex plane.
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