Integral closures and weight functions over finite fields

Leonard, D.A.; Pellikaan, Ruud

doi:10.1109/isit.2002.1023331

Cited by 8 publications

(16 citation statements)

References 13 publications

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“…Problem 42b. In particular, can the Leonard-Pelikaan [86] and Singh-Swanson [104] algorithms be modi…ed for computing the integral closure of ideals?…”

Section: Problem 42mentioning

confidence: 99%

Open Problems in Commutative Ring Theory

Cahen¹,

Fontana

Frisch

et al. 2014

Commutative Algebra

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This article consists of a collection of open problems in commutative algebra. The collection covers a wide range of topics from both Noetherian and non-Noetherian ring theory and exhibits a variety of research approaches, including the use of homological algebra, ring theoretic methods, and star and semistar operation techniques. The problems were contributed by the authors and editors of this volume, as well as other researchers in the area.

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“…Problem 42b. In particular, can the Leonard-Pelikaan [86] and Singh-Swanson [104] algorithms be modi…ed for computing the integral closure of ideals?…”

Section: Problem 42mentioning

confidence: 99%

Open Problems in Commutative Ring Theory

Cahen¹,

Fontana

Frisch

et al. 2014

Commutative Algebra

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“…These papers make use of the Frobenius endomorphism along with a weighted total-degree monomial ordering; this is a monomial ordering under which there are only finitely many elements preceding any given element, and this is an essential ingredient in proving the convergence of their algorithm. The affine domains considered in [Leonard and Pellikaan 2003] are constructed as towers in the following sense: R 0 is a finite field; if R j−1 is given with a weight function wt j−1 , then R j is the integral closure of…”

Section: Remarks and Open Questionsmentioning

confidence: 99%

“…We are very grateful to Douglas Leonard for drawing our attention to [Leonard and Pellikaan 2003] and answering several questions, to David Eisenbud, Ruud Pellikaan, and Wolmer Vasconcelos for their feedback, and to Amelia Taylor for valuable discussions and help with Macaulay 2.…”

Section: Acknowledgementmentioning

confidence: 99%

“…We present an algorithm for computing the integral closure of a reduced ring that is finitely generated over a finite field. 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;Stolzenberg 1968;Vasconcelos 1991;; 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.…”

Section: Anurag K Singh and Irena Swansonmentioning

confidence: 99%

“…Previous algorithms proceed by building successively larger rings between the original ring and its integral closure [ de Jong 1998;Seidenberg 1970;Stolzenberg 1968;Vasconcelos 1991;; 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.…”

Section: Anurag K Singh and Irena Swansonmentioning

confidence: 99%

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2009

ANT

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We introduce T-adic exponential sums associated to a Laurent polynomial f. They interpolate all classical p m-power order exponential sums associated to f. We establish the Hodge bound for the Newton polygon of L-functions of T-adic exponential sums. This bound enables us to determine, for all m, the Newton polygons of L-functions of p m-power order exponential sums associated to an f that is ordinary for m = 1. We also study deeper properties of L-functions of T-adic exponential sums. Along the way, we discuss new open problems about the T-adic exponential sum itself.

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Normalization of rings

Greuel

Laplagne

Seelisch

2010

Journal of Symbolic Computation

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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...

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Integral closures and weight functions over finite fields

Cited by 8 publications

References 13 publications

Open Problems in Commutative Ring Theory

Open Problems in Commutative Ring Theory

Untitled

Normalization of rings

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