Algorithms for group actions in arbitrary characteristic and a problem in singularity theory

TL;DR

This paper develops algorithms to analyze group actions on matrix spaces over formal power series rings in arbitrary characteristic, focusing on finite determinacy, tangent images, and their implications for singularity classification.

Contribution

It introduces algorithms for checking finite determinacy, computing bounds, and analyzing tangent images, highlighting differences in positive characteristic fields.

Findings

Algorithms for finite determinacy and tangent image computation.

In positive characteristic, tangent image and tangent space may differ.

Applications to singularity classification in arbitrary characteristic.

Abstract

We consider the actions of different groups G on the space M of m x n matrices with entries in the formal power series ring K[[x1,..., xs]], K an arbitrary field. G acts on M by analytic change of coordinates, combined with the multiplication by invertible matrices from the left, the right or from both sides, respectively. This includes right and contact equivalence of functions and mappings, resp. ideals. A is called finitely G-determined if any matrix B, with entries of A-B in <x1,...,xs>^k for some k, is contained in the G-orbit of A. The purpose of this paper is to present algorithms for checking finite determinacy, to compute determinacy bounds and to compute the image of the tangent map to the orbit map G -> GA, which we call the tangent image to the orbit GA. The tangent image is contained in the tangent space to the orbit GA and we apply the algorithms to prove that both spaces may be different if the field K has positive characteristic, even for contact equivalence of functions. This fact had been overlooked by several authors before. Besides this application, the algorithms of this paper may be of interest for the classification of singularities in arbitrary characteristic.