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Computing the normalization

The idea for computing the normalization of is as follows:

• We construct an increasing sequence of rings

with a test ideal for , , and a test ideal for , .

• If then .
For performance reasons we do not look for a non-zerodivisor in but choose any non-zero element . If is a zerodivisor then it gives a splitting of the ring which makes the subsequent computations easier.

We obtain the following (highly recursive) algorithm for computing the normalization:

Input
,
We assume that is a radical ideal.
Output
Polynomial rings , ideals , and maps such that , induced by is the normalization map of .

1. Compute an ideal describing the singular locus.