On the Applications of Multi-Equational Resultants

Abstract

Computational methods for manipulating sets of polynomial equations are becoming of greater importance due to the use of polynomial equations in geometric modeling. Recently, the technique of Grabner bases has received much attention as an algorithmic method for determiuing properties of systems of polynomial equations. Grebner bases provide a method for testing ideal membership, a problem whose solution requires running time double exponential in the number of indeterminates (in the worst case). Another lesser known technique based on classical algebraic geometry is that of multi-equational resultants. Computing the resultant of several equations can be done in time single exponential in the number of indeterminates of the equations. In this paper, we survey a range of geometric and algebra.ic problems that may be solved using multi-equational resultants. These problems include converting from the parametric form to the implicit form of curves and surfaces, computing the intersection of three or more surfaces, and computing the convolution of algebraic curves and surfaces. 'Ve also review a method using multi-equational resultants for decomposing an algebraic set into its irreducible components. Finally, we give an original method for computing the image of a hypersurface under a rational map and inverting this map if its is one-t(X)ne. ·Supportetl in part by NSF grant MIP 85.'21356, ARO conlract DAAG2g·S5-C-001S under Cornell l1SI and ONR conlract NOOOH-SS-K-040'2 tSupported in parl by NSF grant IRI 8S-10i47