Let G be a finite matrix group and X be a Gvariety.
We propose a new approach for computing a stratification of X with respect to
the orbit type, respectively
of the quotient X/G and we present new algorithms for this task. For X = R^n
these algorithms yield an optimal
description of each stratum and of the orbit space in terms of polynomial
equations and inequalities
(optimal with respect to the number of inequalities). Moreover we show that the
dimension d of a stratum S is an upper and lower bound for the number of
inequalities needed for a description of S
and its closure, which improves the upper bound d(d+1)/2, which holds for
general basic closed semialgebraic sets of
dimension d.
Additionally, our algorithms allow to compute strata of particular interest,
which demands
less computational resources. By performing computations as long as possible in
the representation space and by
refining results of Procesi and Schwarz, it seems that our algorithms are more
efficient than the present approach.
We conclude by giving an application of our algorithms to the problem of
constructing a potential for
NickelTitanium alloys and compare the runtime with other algorithms.
