We present an algorithm for computing the integral closure of a reduced ring
that is finitely generated over a finite field

Singh, Anurag K.

Swanson, Irena

English

arXiv

Abstract. We present an algorithm for computing the integral closure of a reduced ring that is finitely generated over a finite field. Leonard and Pellikaan [4] devised an algorithm for computing the integral closure of weighted rings that are finitely generated over finite fields. Pre-vious algorithms proceed by building successively larger rings between the original ring and its integral closure, [2, 6, 7, 9, 11, 12]. The Leonard-Pellikaan algorithm instead starts with the first approximation being a finitely generated module that contains the integral closure, and succes-sive steps produce submodules containing the integral closure. The weights in [4] impose strong restrictions; these weights play a crucial role in all steps of their algorithm. We present a modification of the Leonard-Pellikaan al-gorithm which works in much greater generality: it computes the integral closure of a reduced ring that is finitely generated over a finite field. We discuss an implementation of the algorithm in Macaulay 2, and pro-vide comparisons with de Jong’s algorithm [2]. 1. The algorithm Our main result is the following theorem; see Remark 1.5 for an algorith-mic construction of an element D as below when R is a domain, and for techniques for dealing with the more general case of reduced rings. Theorem 1.1. Let R be a reduced ring that is finitely generated over a computable field of characteristic p&gt; 0. Set R to be the integral closure of R in its total ring of fractions. Suppose D is a nonzerodivisor in the conductor ideal of R, i.e., D is a nonzerodivisor with DR ⊆ R. (1) Set V0 = 1DR, and inductively define Ve+1 = {f ∈ Ve | fp ∈ Ve} for e&gt; 0. Then the modules Ve are algorithmically constructible. (2) The descending chai

An algorithm for computing the integral closure

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