Emilio Carvalho | Searcharxiv

arXiv2020

Standard Bases for Fractional Ideals of the Local Ring of an Algebroid Curve

In this paper we present an algorithm to compute a Standard Basis for a fractional ideal $\mathcal{I}$ of the local ring $\mathcal{O}$ of an $n$-space algebroid curve with several branches. This allows us to determine the semimodule of values of $\mathcal{I}$. When $\mathcal{I}=\mathcal{O}$, we may obtain a (finite) set of generators of the semiring of values of the curve, which determines its classical semigroup. In the complex context, identifying the Kähler differential module $Ω_{\mathcal{O}/\mathbb{C}}$ of a plane curve with a fractional ideal of $\mathcal{O}$ and applying our algorithm, we can compute the set of values of $Ω_{\mathcal{O}/\mathbb{C}}$, which is an important analytic invariant associated to the curve.

math.AG↗

arXiv2018

The Semiring of Values of an Algebroid Curve

We introduce the semiring of values $Γ$ with respect to the tropical operations associated to an algebroid curve. As a set, $Γ$ determines and is determined by the well known semigroup of values $S$ 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 of each branch of the curve and the mutual intersection multiplicity of its branches. In the analytical case, this allows us to connect directly the results of Zariski and Waldi that characterize the topological type of the curve.

math.AG↗