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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.turns out to be very fast for small p. However the computation of the Frobenius makes it impracticable when p is large.There are also very efficient methods for computing the normalization in some special cases. For example, for toric rings, one can apply fast combinatorial techniques, as explained in Bruns and Koch (2001).The algorithm we propose in this paper is a general algorithm and it is based on de Jong (1998) and Decker et al. (1999). In their algorithm, as we mentioned before, they construct an increasing chain of affine rings. They enlarge the rings by computing the endomorphism ring of a test ideal (see below), adding new variables for each module generator of the endomorphism ring and dividing out the relations among them. Then the algorithm is applied recursively to this new affine ring. Due to the increasing number of variables and relations this can produce a big slowdown in the performance of the algorithm already when the number of intermediate rings is 2 or 3. For a larger number, it usually makes the algorithm unusable, as the Groebner bases of the ideals of relations grow extensively. Our approach avoids the increasing complexity when enlarging the rings, benefiting from the finitely generated A-module structure of the normalization. We are able to do most computations over the original ring without adding new variables or relations.The main new results of this paper are presented in Section 3. In Section 4 we describe the algorithm and show, as an application, how the δ-invariant of the ring can be computed. Section 5 contains several benchmark examples and a comparison with previously known algorithms, while Section 6 is devoted to an extension of the algorithm to non-global monomial orderings. Basic definitions and toolsLet A be a reduced Noetherian ring. 1 The normalizationĀ of A is the integral closure of A in the total ring of fractions Q(A), which is the localization of A with respect to the non-zerodivisors on A. A is called normal if A =Ā.The conductor of A inĀ is C = {a ∈ Q(A) | aĀ ⊂ A} = Ann A (Ā/A).Lemma 2.1.Ā is a finitely generated A-module if and only if C contains a nonzerodivisor on A.Proof. If p ∈ C is a non-zerodivisor thenĀ ∼ = pĀ ⊂ A is module-finite over A, since A is Noetherian. Conversely, ifĀ is module-finite over A then any common multiple of the denominators of a finite set of generators is a non-zerodivisor on A contained in C.We recall the Grauert and Remmert criterion of normality.Proposition 2.2. Let A be a Noetherian reduced ring and J ⊂ A an idea...

Abstract. The importance of software grows in all sectors of industry and all aspects of life. Given this high dependability on software, the status of software engineering is less than satisfactory. Accidents, recall actions, and late projects still make the news every day. Many of the software engineering research results do not make it into practice, and thereby the gap between research and practice widens constantly. The reasons for not making it into practice range from isufficient commitment for professionalization of software development on the industrial side, to insufficient consideration for practical scale-up issues on the research side, and a tremendous lack of empirical evidence regarding the benefits and limitations of new software engineering methods and tools on both sides. The major focus of this paper is to motivate the creation of credible evidence which in turn will allow for less risky introduction of new software engineering approaches into practice. In order to overcome this progress hindering lack of evidence, both research and practice have to change their paradigms. Research needs to complement each promising new software engineering approach with credible empirical evidence from in vitro controlled experiments and case studies; industry needs to baseline its current state of the practice quantitatively, and needs to conduct in vitro studies of new approaches in order to identify their benefits and limitations in certain industrial contexts.

We construct an explicit minimal strong Gröbner basis of the ideal of vanishing polynomials in the polynomial ring over Z/m for m ≥ 2. The proof is done in a purely combinatorial way. It is a remarkable fact that the constructed Gröbner basis is independent of the monomial order and that the set of leading terms of the constructed Gröbner basis is unique, up to multiplication by units. We also present a fast algorithm to compute reduced normal forms, and furthermore, we give a recursive algorithm for building a Gröbner basis in Z/m[x1, x2, . . . , xn] along the prime factorization of m. The obtained results are not only of mathematical interest but have immediate applications in formal verification of data paths for microelectronic systems-on-chip.

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