An algorithm for computingthe integral closure

Leonard and Pellikaan [2003] 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 [de Jong 1998; Seidenberg 1970; 1975; Stolzenberg 1968; Vasconcelos 1991; 2000]; 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 [Leonard and Pellikaan 2003] impose strong restrictions, and play a crucial role in various steps of their algorithm; see Remark 1.7. We present a modification of the Leonard–Pellikaan algorithm that 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 [1998].