Details:
Title  Using a bihomogeneous resultant to find the singularities of rational space curves  Author(s)  Ronald N. Goldman, Xiaohong Jia, Xiaoran Shi  Type  Article in Journal  Abstract  We provide a new technique to detect the singularities of rational space curves. Given a rational parametrization of a space curve, we first compute a μbasis for the parametrization. From this μbasis we generate three planar algebraic curves of different bidegrees whose intersection points correspond to the parameters of the singularities. To find these intersection points, we construct a new sparse resultant matrix for these three bivariate polynomials. We then compute the parameter values corresponding to the singularities by applying Gaussian elimination to this resultant matrix. Let ν Q denote the multiplicity of the singular point Q, and let n be the degree of the curve. We find that when ∑ ν Q ⩽ 2 n − 3 , the last nonzero row after Gaussian elimination represents a univariate polynomial whose roots are exactly the parameter values of the singularities with the correct multiplicity. Otherwise the last two nonzero rows represent two bivariate polynomials whose common roots provide the parameter values of the singularities. We also show that if R is this resultant matrix, then size ( R ) − rank ( R ) gives the total multiplicity ∑ ν Q ( ν Q − 1 ) of all the singular points including the infinitely near singular points of a rational space curve and we provide bounds on the expression ∑ ν Q ( ν Q − 1 ) for the total multiplicity of all the singular points of a rational space curve. To verify our results, we present several examples to illustrate our methods.  Keywords  Rational space curve, Resultant matrix, μBasis, Singularities, Intersection number  ISSN  07477171 
URL 
http://www.sciencedirect.com/science/article/pii/S074771711200171X 
Language  English  Journal  Journal of Symbolic Computation  Volume  53  Number  0  Pages  1  25  Year  2013  Edition  0  Translation 
No  Refereed 
No 
