Residual resultant over the projective plane and the implicitization problem

In this article, we first generalize the recent notion of residual resultant of a complete intersection [4] to the case of a local complete intersection of codimension 2 in the projective plane, which is the necessary and sufficient condition for a system of three polynomials to have a solution “outside” a variety, defined here by a local complete intersection of codimension 2. We give its degree in the coefficients of each polynomial and compute it as the god of three polynomials or as a product of two determinants divided by another one. In a second part we use this new type of resultant to give a new method to compute the implicit equation of a rational surface with base points in the case where these base points are a local complete intersection of codimension 2.

[1]  I. M. Gelʹfand,et al.  Discriminants, Resultants, and Multidimensional Determinants , 1994 .

[2]  Bernard Mourrain,et al.  Generalized Resultants over Unirational Algebraic Varieties , 2000, J. Symb. Comput..

[3]  D. Eisenbud.  Commutative Algebra: with a View Toward Algebraic Geometry , 1995 .

[4]  David A. Cox,et al.  Using Algebraic Geometry , 1998 .

[5]  K. Brown,et al.  Graduate Texts in Mathematics , 1982 .

[6]  D. Buchsbaum,et al.  What annihilates a module , 1977 .

[7]  Dinesh Manocha,et al.  A new approach for surface intersection , 1991, SMA '91.

[8]  Dinesh Manocha,et al.  Algorithm for implicitizing rational parametric surfaces , 1992, Comput. Aided Geom. Des..

[9]  Joe W. Harris,et al.  Principles of Algebraic Geometry , 1978 .

[10]  B. Ulrich,et al.  Residual intersections. , 1988 .

[11]  John F. Canny,et al.  Generalised Characteristic Polynomials , 1990, J. Symb. Comput..

[12]  Ron Goldman,et al.  On the Validity of Implicitization by Moving Quadrics for Rational Surfaces with No Base Points , 2000, J. Symb. Comput..

[13]  Christoph M. Hoffmann,et al.  Geometric and Solid Modeling: An Introduction , 1989 .

[14]  Falai Chen,et al.  Implicitization using moving curves and surfaces , 1995, SIGGRAPH.

[15]  B. Mourrain,et al.  Resultant over the residual of a complete intersection , 2001 .

[16]  Marc Chardin.  The Resultant via a Koszul Complex , 1993 .

[17]  David A. Cox,et al.  Ideals, Varieties, and Algorithms , 1997 .

[18]  Dinesh Manocha,et al.  Implicit Representation of Rational Parametric Surfaces , 1992, J. Symb. Comput..

[19]  Carlos D'Andrea.  Resultants and Moving Surfaces , 2001, J. Symb. Comput..