An algorithm for computing the integral closure
Anurag Kumar Singh, Irena Swanson · Algebra & Number Theory · 2009
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. Previous 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 successive 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 algorithm 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 provide comparisons with de Jong’s algorithm [2]. 1. The algorithm Our main result is the following theorem; see Remark 1.5 for an algorithmic 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> 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 = 1 DR, and inductively define Ve+1 = {f ∈ Ve | f p ∈ Ve} for e � 0. Then the modules Ve are algorithmically constructible. (2) The descending chain