29 - Inversion of Birational Maps with Gröbner Bases

Summary

Abstract

We present an efficient algorithm for the inversion of a given birational map. The problem is reduced to finding the unique solution of a maximal ideal defined over an algebraic function field.

Introduction

The problem of inverting a birational maps arises in several contexts in algorithmic algebraic geometry, and an efficient solution is useful in many situations.

For instance, consider the parameterization problem, which is useful for numerous applications in CAD/CAM (see section 3). A closer look to the existing parameterization algorithms reveals that a parameterization is often obtained via inversion of some birational map.

In this paper, we present a new efficient algorithm for the inversion of birational maps, based on the method of Gröbner bases (Buchberger 1965, Buchberger 1979, Buchberger 1983, Beckers and Weispfenning 1993).

For a special case, the inversion problem has also been investigated in (Essen 1990, Audoly et al. 1991) (see also (Ollivier 1989) for related work), in the context of the Jacobian conjecture (Keller 1939). The general problem was solved in (Sweedler 1993) (see also Shannon and Sweedler 1988). However, Sweedler's method depends on computing the components of the inverse map one by one, i.e. it requires the computation of a Gröbner basis with lexicographical termorder. It is known (see Faugère et al. 1993) that such a Gröbner basis is much harder to compute than Gröbner bases with respect to other term orders (e.g. the reverse lexicographical termorder).