An improved algorithm for the resolution of singularities

Bodnár, Gábor; Schicho, Josef

doi:10.1145/345542.345570

Cited by 3 publications

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“…But sometimes one could find an example which seems "easy" but it requires hard middle computations, since coefficients ideals are complicated. The new version [9] has the option to use the feature of Singular for computing in some localizations rings which improves the performance. We will prove resolution of basic objects by induction on the dimension of the ambient space: To find a resolution of a d-dimensional basic object (W, (J, b), E) we will associate to it, at least locally at each point of W , a (d − 1)-dimensional basic object (Z, (A, e), G) in such a way that a resolution of (Z, (A, e), G), is equivalent to a resolution of (W, (J, b), E) in some sense that we will make precise in the forthcoming sections.…”

Section: Bodnár-schicho's Computer Implementationmentioning

confidence: 99%

“…Sing(J, 9) = (H 1 ∩ H 2 ) ∪ (H 1 ∩ H 3 ∩ H 4 ) , and the function h is given by h(ξ) = (−2, 10 9 , (1, 2, 0, 0)) if ξ ∈ H 1 ∩ H 2 (−3, 10 9 , (1, 3, 4, 0)) if ξ ∈ (H 1 ∩ H 3 ∩ H 4 ) \ (H 1 ∩ H 2 ) , so that max h = (−2, 10 9 , (1, 2, 0, 0)) and Max h = H 1 ∩ H 2 . Note that the two irreducible components of Sing(J, 9) The function h 1 corresponding to the basic object (W 1 , (J 1 , 9), E 1 ) is given as follows:…”

Section: Now One Can Check That the Function H Is Upper-semi-continuomentioning

confidence: 99%

Section: Bodnár-schicho's Computer Implementationmentioning

confidence: 99%

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A Simplified Proof of Desingularization and Applications

Bravo¹,

Encinas²,

Villamayor³

2005

Rev. Mat. Iberoam.

144

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This paper contains a short and simplified proof of desingularization over fields of characteristic zero, together with various applications to other problems in algebraic geometry (among others, the study of the behavior of desingularization of families of embedded schemes, and a formulation of desingularization which is stronger than Hironaka's). Our proof avoids the use of the Hilbert-Samuel function and Hironaka's notion of normal flatness: First we define a procedure for principalization of ideals (i. e. a procedure to make an ideal invertible), and then we show that desingularization of a closed subscheme X is achieved by using the procedure of principalization for the ideal I(X) associated to the embedded scheme X. The paper intends to be an introduction to the subject, focused on the motivation of ideas used in this new approach, and particularly on applications, some of which do not follow from Hironaka's proof.Part 3. Applications. 21 7. Weak and strict transforms of ideals: Strong Factorizing Desingularization. 21 8. On a class of regular schemes and on real and complex analytic spaces. 31 9. Non-embedded desingularization. 34 10. Equiresolution. Families of schemes. 37 11. Bodnár-Schicho's computer implementation. 43Example 3.3. If X ⊂ W is a hypersurface and J is its defining ideal, then Sing(J, b) is the set of points of X where the multiplicity is greater than or equal to b.Example 3.4. If J ⊂ O W is an arbitrary non-zero sheaf of ideals, then Sing(J, b) is the set of points of W where the order of J is greater than or equal to b.for the pair (W, E) and Y ⊂ Sing(J, b). 3.6. Permissible transformations of basic objects. Let (W, (J, b), E = {H 1 , . . . , H r }) be a basic object and let Y ⊂ Sing(J, b) be a permissible center. Consider W ←− W 1 the monoidal transformation with center Y . This induces a transformation of pairs,(3.6.1)as the permissible transformation of the basic object (W, (J, b), E).Remark 3.7. In general, given a sequence of transformations of basic objects (3.7.1)at centers Y i ⊂ Sing(J i , b), i = 0, 1, . . . k − 1, we obtain expressionsNote here that c r+1 = . . . = c r+i = b if none of the centers Y i are included in any of the exceptional divisors H j .Definition 3.8. A finite sequence transformation of basic objects as (3.7.1) is a resolution of (W 0 ,Remark 3.9. Note that:(1) If sequence (3.7.1) is a resolution of the basic object (W 0 ,where M k is an invertible sheaf of ideals, and J k has no points of order ≥ b in W k . (2) The ideal J k is not the strict transform of J 0 , an ideal which is far more complicated to define (see Section 7 for a discussion on this matter). However it is so in some particular cases. In fact, if X 0 ⊂ W 0 is a closed smooth subscheme, and J 0 = I(X), then Sing(J 0 , 1) = X, and given any sequence of transformations of basic objects

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Section: Bodnár-schicho's Computer Implementationmentioning

confidence: 99%

Section: Now One Can Check That the Function H Is Upper-semi-continuomentioning

confidence: 99%

See 1 more Smart Citation

A Simplified Proof of Desingularization and Applications

Bravo¹,

Encinas²,

Villamayor³

2005

Rev. Mat. Iberoam.

144

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“…In this paper we present generalized and corrected results of our contribution to ISSAC 2000 (Bodnár and Schicho, 2000c), where we discussed three improvements of Villamayor's algorithm for resolution of singularities. This time we omit the description of the third improvement, which describes a sophisticated representation of resolution problems, because we think it is so closely related to the algorithm that it is not really interesting for a larger audience.…”

Section: Introductionmentioning

confidence: 99%

Two Computational Techniques for Singularity Resolution

Bodnár

Schicho²

2001

Journal of Symbolic Computation

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This paper presents two efficient computational techniques in algebraic geometry. The first one allows the elimination of redundancies in the representation of quasi-projective varieties by atlases of affine charts. The second simplifies the computations with exponentiated ideals by attaching rational weights to the generators, applying Hironaka's theory of idealistic exponents. As the main application, we used these techniques to speed up Villamayor's algorithm for resolving hypersurface singularities in any dimension.

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Algorithmic Tests for the Normal Crossing Property

Bodnár

2004

Automated Deduction in Geometry

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An improved algorithm for the resolution of singularities

Cited by 3 publications

References 11 publications

A Simplified Proof of Desingularization and Applications

A Simplified Proof of Desingularization and Applications

Two Computational Techniques for Singularity Resolution

Algorithmic Tests for the Normal Crossing Property

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