Revista Matemática Iberoamericana


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Volume 15, Issue 3, 1999, pp. 621–634
DOI: 10.4171/RMI/267

Published online: 1999-12-31

On proximity relations for valuations dominating a two-dimensional regular local ring

José Juan Aparicio[1], Ángel Granja[2] and Tomás Sánchez-Giralda[3]

(1) Universidad de Valladolid, Spain
(2) Universidad de León, Spain
(3) Universidad de Valladolid, Spain

The purpose of this paper is to define a new numerical invariant of valuations centered in a regular two-dimensional regular local ring. For this, we define a sequence of non-negative rational numbers $\delta_\nu = \lbrace \delta_\nu (j)\rbrace_{j≥0}$ which is determined by the proximity relations of the successive quadratic transformations at the points determined by a valuation $\nu$. This sequence is characterized by seven combinatorial properties so that any sequence of non-negative rational numbers having the above properties is the sequence associated to a valuation.

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Aparicio José Juan, Granja Ángel, Sánchez-Giralda Tomás: On proximity relations for valuations dominating a two-dimensional regular local ring. Rev. Mat. Iberoamericana 15 (1999), 621-634. doi: 10.4171/RMI/267