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Original Articles

The value semiring of an algebroid curve

Pages 3275-3284 | Received 24 Jun 2019, Accepted 18 Feb 2020, Published online: 09 Mar 2020
 

Abstract

We study the value semiring Γ, equipped with the tropical operations, associated to an algebroid curve. As a set, Γ determines and is determined by the well-known value semigroup S and we prove that Γ is always finitely generated in contrast to S. In particular, for a plane curve, we present a straightforward way to obtain Γ 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.

2010 MATHEMATICS SUBJECT CLASSIFICATION:

Acknowledgments

The authors are grateful to the anonymous referee for the suggestions that improve this work.

Additional information

Funding

The first author was partially supported by CNPq and the second one by CAPES.

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