2023/01/03 by João T. Cabral, Cabral, Joao, Ana Palmira Bittencourt Santos Casimiro +1
Computer Science · Engineering · #Advanced Numerical Analysis Techniques #Algebraic Geometry (math.AG) #FOS: Mathematics #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.2301.01155
openalex publication_date 2023/01/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 \citeCM) 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 Gröbner bases.