## 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, old)
Mathematica 3.0 Version (also Intersection2, called SubAlgIntersect, old)
Singular Version (new)
Singular Examples (new)

## 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)

Singular Version
Singular Examples (8.11.2002)

## Singularity Theory

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.

### The rinvar library

The zeroset library

## 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.
1. Compute the orbit of points, varieties, and generic points.
2. Compute the stabilizer of points and of G-varieties.
3. Compute the principal (generic) stabilizer (decide if an action is free).
4. Compute a stratification of the representation space of G.
5. Compute a description of (all or some particular) strata, repsectively their closures of the orbit space, which is optimal in the number of inequalities.
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. Strata.sing and an Example

## Algorithmic Combinatorics

In a programming project by Dr. Peter Paule (RISC-Linz, combinatorics), I have implemented the Mathematica package PermGroup for Polya theory and permutation groups.