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

Abstract

Let ${M}_{m,n}$ denote the space of $m\times n$ matrices with entries in the formal power series ring $K[[x_1,\ldots , x_s]], K$ an arbitrary field. We consider different groups G acting on $M_{m,n}$ by formal change of coordinates, combined with the multiplication by invertible matrices. This includes right and contact equivalence of functions, mappings, and ideals. A matrix A is called finitely G-determined if any matrix B, with entries of $A-B$ in $\langle x_1,\ldots ,x_s\rangle ^k$ for some k, is contained in the G-orbit of A. In this paper we present algorithms to check finite determinacy, to compute determinacy bounds and to compute the tangent image ${\widetilde{T}}_A(GA)$ of the action. The tangent image is an important invariant in positive characteristic since it differs in general from the tangent space $T_A(GA)$ to the orbit of G (in a subtle way). This fact was only recently discovered by the authors and is proved in the present paper by using our algorithms. Besides this application, the algorithms of this paper are of interest for the classification of singularities in arbitrary characteristic, a subject of growing interest.