A generalization of Gröbner bases helps to compute singularity theory transformations

Abstract

According to singularity theory, many functions admit (local) normal forms under suitable equivalence transformations. Motivated by a dynamical systems problem, where we are interested in symbolically computing bifurcation curves, our goal is to compute the normalizing transformation explicitly. It's computation proceeds stepwise, and resembles Newton's root-finding algorithm, with the derivative replaced by the tangent space to the orbit under equivalence transformations. At each step we need a solution of a linear equation involving the tangent space, analogous to Newton's algorithm requiring the inverse of the derivative. If the equivalences are right-transformations, the tangent space is an ideal, and the linear equation is solved by the normal form or 'division' algorithm of Grobner bases. If left-right transformations are used, the tangent space consists of sums of ideal and algebra elements. The resulting linear equation can be solved analogously by suitably extending the notions of Groebner basis and canonical subalgebra (SAGBI) basis. See [9] for an extended overview, [8] for the details.