# The valuative tree is the projective limit of Eggers-Wall trees

1. González Pérez, Pedro D.
2. García Barroso, Evelia R.
3. Popescu-Pampu, Patrick
1. 1 Universidad de La Laguna
info

San Cristobal de La Laguna, España

info

ROR 02p0gd045

Revista:
Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas

ISSN: 1578-7303 1579-1505

Any de publicació: 2019

Volum: 113

Número: 4

Pàgines: 4051-4105

Tipus: Article

Altres publicacions en: Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas

## Resum

Consider a germ C of reduced curve on a smooth germ S of complex analytic surface. Assume that C contains a smooth branch L. Using the Newton-Puiseux series of C relative to any coordinate system (x, y) on S such that L is the y-axis, one may define the Eggers-Wall treeΘL(C) of C relative to L. Its ends are labeled by the branches of C and it is endowed with three natural functions measuring the characteristic exponents of the previous Newton-Puiseux series, their denominators and contact orders. The main objective of this paper is to embed canonically ΘL(C) into Favre and Jonsson’s valuative tree P(V) of real-valued semivaluations of S up to scalar multiplication, and to show that this embedding identifies the three natural functions on ΘL(C) as pullbacks of other naturally defined functions on P(V). As a consequence, we generalize the well-known inversion theorem for one branch: if L′ is a second smooth branch of C, then the valuative embeddings of the Eggers-Wall trees ΘL′(C) and ΘL(C) identify them canonically, their associated triples of functions being easily expressible in terms of each other. We prove also that the space P(V) is the projective limit of Eggers-Wall trees over all choices of curves C. As a supplementary result, we explain how to pass from ΘL(C) to an associated splice diagram.

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