Implicitization Of A Plane Curve Germ

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.