Two New Algorithms in Algebraic Geometry

We report about two new algorithms, "Normalization of affine rings" and "Monodromy of isolated hypersurface singularities".

The normalization algorithm relies on a criterion of Grauert and Remmert for normality from "Analytische Stellenalgebren", Springer 1971, which seem to have been escaped the notice of the computer algebra community. Its transformation into an algorithm uses basic standard basis computations and has turned out to be quite effective. We shall report about the theory, the implementation and the experiments.

The algorithm for computing the monodromy had been proposed by Brieskorn in 1971. Its realization, together with an extension and some refinements, requires standard basis computations in local rings and nontrivial reduction to solving (big) systems of linear equations.

Both algorithms have been recently implemented in SINGULAR and we shall demonstrate a few examples. Participants are encouraged to provide their own examples for testing the algorithms at the end of the talk.

References:

W. Decker, G.-M. Greuel, G. Pfister, T. De Jong:
The normalisation: a new algorithm, implementation and comparisons.
In: Proc. EUROCONFERENCE Computational Methods for Representations of
Groups and Algebras (1.4.-5.4.1997), Birkhauser 1998. 

M. Schulze:
Computation of the Monodromy of the Meromorphic Gauss-Manin
Connection. 
Diplomarbeit, Kaiserslautern 1999.