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

Abstract

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 ${\delta }_{\nu }=\{{\delta }_{\nu }(j){\}}_{j\ge 0}$ 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.