Normalization of Rings

TL;DR

This paper introduces a novel, faster algorithm for computing the integral closure of reduced Noetherian rings, with an efficient modification for positive characteristic cases, implemented in Singular.

Contribution

It presents a new normalization algorithm for rings, including a positive characteristic modification, with implementation and benchmarking demonstrating superior speed.

Findings

The algorithm is significantly faster than existing methods.

Implementation in Singular enables practical computations.

Benchmark tests confirm improved performance.

Abstract

We present a new algorithm to compute the integral closure of a reduced Noetherian ring in its total ring of fractions. A modification, applicable in positive characteristic, where actually all computations are over the original ring, is also described. The new algorithm of this paper has been implemented in Singular, for localizations of affine rings with respect to arbitrary monomial orderings. Benchmark tests show that it is in general much faster than any other implementation of normalization algorithms known to us.