Using a Bivariate Sparse Resultant to Find the Singularities of Rational Space Curves
Xiaoran Shi, University of Science and Technology of China/Rice

Singularities are the most interesting and important points on curves and surfaces. We want to detect and analyze the singularities of rational space curves. The problem of computing the singularities can be reduced to the problem of computing the intersection points of three related planar algebraic curves. To analyze the intersections of these three planar curves, we construct a new sparse resultant matrix and derive a relationship between the kernel of this resultant matrix and the parameters of the singularities. We then find the singularities of a rational space curve by applying Gaussian Elimination to this sparse resultant matrix. Some examples are presented to illustrate our methods.