Implicitization Of A Plane Curve Germ

TL;DR

This paper introduces an algorithm for implicitizing irreducible plane curve germs that computes polynomials matching the semigroup generators, utilizing tropical geometry and truncations of parametrizations.

Contribution

The authors develop a novel algorithm combining tropical geometry and Puiseux series truncations to implicitly describe plane curve germs, with proven finiteness and complexity analysis.

Findings

Algorithm computes semigroup generator polynomials effectively.

Complexity matches that of an integer linear programming problem.

Implementation shows improved computation times over Gröbner basis elimination.

Abstract

Let $Y=\{f(x,y)=0\}$ be the germ of an irreducible plane curve. We present an algorithm to obtain polynomials, whose valuations coincide with the semigroup generators of $Y$. These polynomials are obtained sequentially, adding terms to the previous one in an appropriate way. To construct this algorithm, we perform truncations of the parametrization of $Y$ induced by the Puiseux Theorem. Then, an implicitization theorem of Tropical Geometry (Theorem $1.1$ of \cite{CM}) for plane curves is applied to the truncations. The identification between the local ring and the semigroup of $Y$ plays a key role in the construction of the algorithm, allowing us to carry out a formal elimination process, which we prove to be finite. The complexity of this elimination process equals the complexity of a integer linear programming problem. This algorithm also allows us to obtain an approximation of the series $f$, with the same multiplicity and characteristic exponents. We present a pseudocode of the algorithm and examples, where we compare computation times between an implementation of our algorithm and elimination through Gr\"{o}bner bases.