Implicitizing rational curves by the method of moving quadrics

International audienceA new technique for finding implicit matrix-based representations of rational curves in arbitrary dimension is introduced. It relies on the use of moving quadrics following curve parameterizations, providing a high-order extension of the implicit matrix representations built from their linear counterparts, the moving planes. The matrices we obtain offer new, more compact, implicit representations of rational curves. Their entries are filled by linear and quadratic forms in the space variables and their ranks drop exactly on the curve. Typically, for a general rational curve of degree d we obtain a matrix whose size is half of the size of the corresponding matrix obtained with the moving planes method. We illustrate the advantages of these new matrices with some examples, including the computation of the singularities of a rational curve

Generators of the ideal of an algebraic space curve

In this paper we show that the ideal of any algebraic curve in affine 3-space whose Jacobian matrix has rank at least 1 at every singular point of the curve can be generated by three polynomials and we give constructive procedures to compute such generators