On the complexity of the integral closure

Ulrich, Bernd; Vasconcelos, Wolmer V.

doi:10.1090/s0002-9947-04-03627-x

Cited by 3 publications

(3 citation statements)

References 27 publications

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“…Let us phrase one of these situations in the context of Corollary 1.3.35. A different and more comprehensive approach that emphasizes exclusively the multiplicity can be found in [185,Section 3].…”

Section: Therefore N ≤ Jdeg(a/a) ✷mentioning

confidence: 99%

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Complexity Degrees of Algebraic Structures

Vasconcelos

2014

Preprint

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We aim at studying collections C(R) of algebraic structures defined over a commutative ring R and investigating the complexity of significant constructions carried out on these objects. Noteworthy are the category of finitely generated modules (over local rings, or graded, particularly vector bundles), finitely generated algebras or objects suitable of more fundamental decompositions and construction of various closure operations-such as the determination of global sections and the (theoretical or machine) computation of integral closures of algebras. Typically, the operations express smoothing processes that enhance the structure of the algebras, enable them to support new constructions (including analytic ones), divisors acquire a group structure, and the cohomology tends to slim down. The assignment of measures of size, via a multiplicity theory, to the algebras and to the construction itself is a novel aspect to the subject. One of its specific goals is to develop a comprehensive theory of normalization in commutative algebra and a broad set of multiplicity functions to be used as complexity benchmarks.

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Section: Therefore N ≤ Jdeg(a/a) ✷mentioning

confidence: 99%

“…For contrast, wed briefly describe some of the related results of [185,Section 3]. Note that their emphasis in exclusively on the multiplicity.…”

Section: Therefore N ≤ Jdeg(a/a) ✷mentioning

confidence: 99%

Complexity Degrees of Algebraic Structures

Vasconcelos

2014

Preprint

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“…Given the usefulness of Theorem 2.5, it would be worthwhile to look at the situation short of Cohen-Macaulayness. For the integral closure of a standard graded algebra A of dimension d, it was possible in [14] to derive degree bounds assuming only S d−1 for A. Another issue is to compute the relation type of A in Theorem 2.5.…”

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confidence: 99%

Indices of normalization of ideals

Polini

Ulrich

Vasconcelos

et al. 2019

Journal of Pure and Applied Algebra

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We derive numerical estimates controlling the intertwined properties of the normalization of an ideal and of the computational complexity of general processes for its construction. In [8], this goal was carried out for equimultiple ideals via the examination of Hilbert functions. Here we add to this picture, in an important case, how certain Hilbert functions provide a description of the locations of the generators of the normalization of ideals of dimension zero. We also present a rare instance of normalization of a class of homogeneous ideals by a single colon operation.

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Complexity of the normalization of algebras

Pham

Vasconcelos

2007

Math. Z.

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We introduce and develop new techniques to study the complexity of normalization processes of graded algebras. The construction of a new degree function on graded modules, with a global nature, permits a broad extension of recent bounds for the length of the chains of subalgebras that general algorithms must transverse to build the integral closure, particularly of blowup algebras. It achieves this by relating the values of the new degree with invariants of the algebra known ab initio. As a by-product, it reveals new inequalities among Hilbert coefficients.

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On the complexity of the integral closure

Cited by 3 publications

References 27 publications

Complexity Degrees of Algebraic Structures

Complexity Degrees of Algebraic Structures

Indices of normalization of ideals

Complexity of the normalization of algebras

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