The value semiring of an algebroid curve

摘要

We study the value semiring Gamma, equipped with the tropical operations, associated to an algebroid curve. As a set, Gamma determines and is determined by the well-known value semigroup S and we prove that Gamma is always finitely generated in contrast to S. In particular, for a plane curve, we present a straightforward way to obtain Gamma in terms of the semiring (or the semigroup) of each branch of the curve and the mutual intersection multiplicity of its branches. In the analytic case, this allows us to relate the results of Zariski and Waldi that characterize the topological type of the curve.