Calculation system for the dual graph of resolution of the algebraic curve singularities using Risa/Asir
Mitsushi Fujimoto
fujimoto@fukuoka-edu.ac.jpEmail Address of #2
Department of Mathematics
Fukuoka University of Education
729-1, Akama, Munakata, Fukuoka 811-41, Japan
Masakazu Suzuki
suzuki@math.kyushu-u.ac.jp
Graduate School of Mathematics
Kyushu University
36, Fukuoka 812, Japan
Abstract
A resolution of an algebraic curve C at a singular point P of C is said "minimal normal", if the union of the proper transform of C and the exceptional curve E of the resolution is of normal crossing type in a neighborhood of E, and it is "minimal" among the resolutions having this property. Usually, the geometric configuration of the resolution is represented by a weighted graph, called the "dual graph", with vertices representing the irreducible components of the exceptional curves joined by a segment each other in case the corresponding components intersect, the vertices having the weght equal to its self-intersection number.
The dual graph of the minimal normal resolution of algebraic curve singularities is one of the most fundamental invariants in algebraic geometry.
Our system computes the dual graphs of the minimal normal resolutions of algebraic curve singularities at origin defined by polynomials of two variables with rational coefficients. The dual graph is calculated from the coefficients of the given polynomial exactly, the computing process including no approximation.
In the algorithm Newton Polygon Algorithm is used to obtain Puiseux series. Newton Polygon Algorithm needs finding roots of polynomials with coefficients in the algebraic extension field over the rational number field. We implemented this algorithm on computer algebra system Risa/Asir.
Users can get the resulting dual graph by the list form and also by the graphical form.
© Asian Technology Conference in Mathematics, 1998. |
