A new class of term orders for elimination

Elimination is a classical subject. The problem is algorithmically solvable by using resultants or by one calculation of Groebner basis with respect to an elimination term order. However, there is no existing method that is both efficient and reliable enough for applicable size problems, say implicitization of bi-cubic Bezier surfaces with degree six in five variables. This basic and useful operation in computer aided geometric design and geometric modeling defies a solution even when approximation using floating-point or modular coefficients is used for Groebner basis computation. An elimination term order can be used to eliminate U for any ideal in K[X][U]. However, for most practical problems we are given a fixed ideal, which means that an elimination term order may be too much for our calculation. In this paper, the author proposes a new approach for elimination. Instead of using a classical elimination term order for all problems or ideals as usual, the author proposes to use algebraic structures of the given system of equations for finding more suitable term orders for elimination of the given problem only. Experimental results showed that these ideal-specific term orders are much more efficient for elimination. In particular, when ideal specific term orders for elimination are used with Groebner walk conversion, one can completely avoid all perturbations. This is a significant result because researchers have been struggling with how to perturb basis conversions for a long time.