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00925njm 22002777a 4500 dc García Barroso, Evelia R. author González Pérez, Pedro Daniel author Popescu Pampu, Patrick author 2019-03-08 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 tree of ΘL (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 (ν) 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 (ν). 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 (ν) 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. García Barroso, E.R., González Pérez, P.D., Popescu-Pampu, P.: The valuative tree is the projective limit of Eggers-Wall trees. RACSAM. 113, 4051-4105 (2019). https://doi.org/10.1007/s13398-019-00646-z 10.1007/s13398-019-00646-z https://hdl.handle.net/20.500.14352/92228 https//doi.org/10.1007/s13398-019-00646-z https://link.springer.com/article/10.1007/s13398-019-00646-z The valuative tree is the projective limit of Eggers-Wall trees