Packages for Computer Algebra Systems
- An Algorithm for Computing the Intersection of Invariant Rings
- An Algorithm for Computing Invariants up to a given Degree
- Singularity Theory (Singular libraries)
- Computing Stratifications of Actions of Compact Lie Groups
- Algorithmic Combinatorics
An Algorithm for Computing the Intersection of Invariant Rings
In my thesis, written at RISC-Linz,
I have developed two algorithms for the computation of the intersection
of invariant rings R, S, provided that R, S, and R intersected with S are
finitely generated. A detailed description of the first algorithm can be
found in my diploma thesis Algorithmic
Aspects of Invariant Theory. The second algorithm, together with a
refined version of the first algorithm, is contained in my paper Computing
the Intersection of Invariant Rings.
The Intersection Algorithm
Magma Version (Intersection1,
3.0 Version (also Intersection2, called SubAlgIntersect, old)
An Algorithm for Computing Invariants up to a given Degree
The invariant ring of the linear action of an algebraic group G can be
expressed as the intersection of two subrings of the coordinate ring of
the variety Gx K^n. Since this ring is a quotient ring one cannot use the
algorithm for computing intersections of subrings of the polynomial ring (see
An Algorithm for Computing Invariants of Linear Actions of Algebraic Groups up to a given Degree )
Note: This algorithm is not restricted to reductive groups,
in particular, it can be used to compute invariants up to a given degree
of unipotent groups.
Important: The previous version (31.10.2000) contains a bug,
use the new one !
The Algorithm "Invariants" (6.11.2000)
The zeroset library
In my diploma thesis Computation
of moduli spaces for semiquasihomogenous singularities and an implemenation
in SINGULAR, written at the Arbeitsgruppe
Algebraische Geometrie, Computeralgebra und Singularitätentheorie,Fachbereich
Mathematik, University of Kaiserslautern, 2000 (supervised by Prof.
G.-M. Greuel), I have implemented a SINGULAR
library for the computation of such moduli spaces. For a description I
refer to my thesis. These libraries are included in the standard distribution of SINGULAR.
Computing Stratifications of Actions of Compact Lie Groups
We present the package stratify for the computer algebra system SINGULAR V2.0, which
is based on our work on the stratification of group actions of compact Lie groups (see
Computing Stratifications of Compact Lie Groups ) and specialized
algorithms for finite groups (see
Optimal Descriptions of Orbit Spaces and Strata of finite Groups).
The package is described in
STRATIFY - A SINGULAR Package for Computing Stratifications of Compact Group Actions
More precisely, the package contains algorithms for the following problems.
Contrary to the approaches of Abud-Sartori and Gatermann, our algorithms construct a stratification of the
representation space of G, and only then to construct the stratification of the orbit space (or the images of
relevant strata) by means of elimination theory (equations) and refinements of results of Procesi and Schwarz
(inequalities). Our approach has several advantages compared to the present approach
namely: Primary decomposition is done before the (nonlinear) Hilbert map is applied, no superfluous
components in the orbit space are computed, the association of strata and their stabilizers is quite obvious, and
it is possible to computpe descriptions of individual strata relevant for applications.
Our algorithms describe any d-dimensional stratum and its closure by at most d inequalities, which turns out to
be optimal. In addition, they allow the computation of a d-dimensional stratum up to generic equivalence by means of
d inequalitites with fewer terms.
For several applications, like the construction of continuous potentials on the orbit space, this approach may lead
to easier computations. For polynomial potentials, inequalities need not be calculated since the Zariski-closure of
a stratum suffices.
and an Example
- Compute the orbit of points, varieties, and generic points.
- Compute the stabilizer of points and of G-varieties.
- Compute the principal (generic) stabilizer (decide if an action is free).
- Compute a stratification of the representation space of G.
- Compute a description of (all or some particular) strata, repsectively their closures
of the orbit space, which is optimal in the number of inequalities.
In a programming project by Dr. Peter Paule (RISC-Linz,
I have implemented the Mathematica
package PermGroup for Polya theory and