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 which is determined by the proximity relations of the successive quadratic transformations at the points determined by a valuation . 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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José Juan Aparicio, Ángel Granja, Tomás Sánchez-Giralda, On proximity relations for valuations dominating a two-dimensional regular local ring. Rev. Mat. Iberoam. 15 (1999), no. 3, pp. 621–634DOI 10.4171/RMI/267