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We consider the actions of different groups G on the space M of m x nmatrices with entries in the formal power series ring K[[x1,..., xs]], K anarbitrary field. G acts on M by analytic change of coordinates, combined withthe multiplication by invertible matrices from the left, the right or from bothsides, respectively. This includes right and contact equivalence of functionsand 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 ofA. The purpose of this paper is to present algorithms for checking finitedeterminacy, to compute determinacy bounds and to compute the image of thetangent map to the orbit map G -> GA, which we call the tangent image to theorbit GA. The tangent image is contained in the tangent space to the orbit GAand we apply the algorithms to prove that both spaces may be different if thefield K has positive characteristic, even for contact equivalence of functions.This fact had been overlooked by several authors before. Besides thisapplication, the algorithms of this paper may be of interest for theclassification of singularities in arbitrary characteristic.

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