Test ideals in diagonal hypersurface rings, II

Let R=k[x1,…,xn]/(x1d+ +xnd), where k is a field of characteristic p, p does not divide d and n 3. We describe a method for computing the test ideal for these diagonal hypersurface rings. This method involves using a characterization of test ideals in Gorenstein rings as well as developing a way to compute tight closures of certain ideals despite the lack of a general algorithm. In addition, we compute examples of test ideals in diagonal hypersurface rings of small characteristic (relative to d) including several that are not integrally closed. These examples provide a negative answer to Smithʹs question [K.E. Smith, The multiplier ideal is universal test ideal, Comm. Algebra 28 (12) (2000) 5912–5929] of whether the test ideal in general is always integrally closed.