An in depth analysis, via resultants, of the singularities of a parametric curve. (English) Zbl 1439.14171
The knowledge of the singular locus of an algebraic curve is essential for a variety of algorithms.
Given a defining polynomial \(p(x,y)\) for the curve, we simply have to determine the common zeros of \(p\) and its partial derivatives.
The situation is more complicated if the curve is given by a rational parametrization; i.e., a pair of rational functions s.t. the curve is the Zariski closure of the image.
In this paper a so-called T-function is determined by means of a resultant computation on the parametrization. By factorizing this T-function one can obtain the singularities (ordinary or non-ordinary) together with their multiplicities.
In this paper a so-called T-function is determined by means of a resultant computation on the parametrization. By factorizing this T-function one can obtain the singularities (ordinary or non-ordinary) together with their multiplicities.
Reviewer: Franz Winkler (Linz)
MSC:
| 14Q05 | Computational aspects of algebraic curves |
| 65D18 | Numerical aspects of computer graphics, image analysis, and computational geometry |
| 14Q65 | Geometric aspects of numerical algebraic geometry |
Keywords:
rational parametrization; singularities of an algebraic curve; multiplicity of a point; ordinary and non-ordinary singularities; T-function; fiber functionSoftware:
CASAReferences:
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