NDLI: Optimal descriptions of orbit spaces and strata of finite groups

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Optimal descriptions of orbit spaces and strata of finite groups

Content Provider	ACM Digital Library
Author	Bayer, Thomas
Abstract	Let G ⊆ $GL;_{n;}$ (R be a finite matrix group and X⊆R $^{n}$ be a G-variety. We propose a new approach for computing a stratification of X with respect to the orbit type of $R^{n}$ 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 ∑ $;_{d;}$ of $R;_{n;}/G$ is an upper and lower bound for the number of inequalities needed for a description of ∑ $;_{d;}$ 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 of X/G, which demands less computational resources. By performing computations as long as possible in $R^{n}$ (and not in $R^{n}/G)$ 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 Nickel-Titanium alloys and compare the runtime with other algorithms.
Starting Page	19
Ending Page	26
Page Count	8
File Format	PDF
ISBN	158113827X
DOI	10.1145/1005285.1005291
Language	English
Publisher	Association for Computing Machinery (ACM)
Publisher Date	2004-07-04
Publisher Place	New York
Access Restriction	Subscribed
Subject Keyword	Stratification Symbolic computation
Content Type	Text
Resource Type	Article

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