Effective computation of base points of ideals in two-dimensional local rings

Alberich-Carramiñana, Maria; Montaner, Josep Àlvarez; Blanco, Guillem

doi:10.1016/j.jsc.2018.01.001

Cited by 4 publications

(7 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…One can compute the logresolution divisor of a using the algorithm from [2]. The log-resolution and its associated divisor F of a are precisely the proper birational morphism π and the divisor D from Example 2.8.…”

Section: 1mentioning

confidence: 99%

“…Combining the algorithms developed in [2] and [5] with Algorithm 3.5 we may provide a method that, given a set of generators of a planar ideal a, returns the set of jumping numbers and a set of generators of the corresponding multiplier ideals. Namely, we have to perform the following steps:…”

Section: Multiplier Idealsmentioning

confidence: 99%

“…The logresolution of a can be computed using the algorithm from [2] and is represented by means of the following dual-graph: The divisor F such that a·O ′…”

Section: Multiplier Idealsmentioning

confidence: 99%

“…Then, the integral closure a is the ideal H D . We should point out that, given a set of generators of the ideal a we may compute the divisor D using the algorithm that we developed in [2].…”

Section: Introductionmentioning

confidence: 99%

“…This is the case of multiplier ideals which is an invariant of singularities that has received a lot of attention in recent years. Combining the algorithms given in [2] and [5] with Algorithm 3.5 we develop a method that, given a set of generators of a planar ideal, returns a set of generators of the corresponding multiplier ideals. Since multiplier ideals are invariant up to integral closure, we obtain a result that resembles a formula given by Howald [18] in the sense that the multiplier ideals of a monomial ideal (in the set of maximal contact elements) is monomial as well.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Monomial generators of complete planar ideals

2020

Self Cite

View full text Add to dashboard Cite

We provide an algorithm that computes a set of generators for any complete ideal in a smooth complex surface. More interestingly, these generators admit a presentation as monomials in a set of maximal contact elements associated to the minimal log-resolution of the ideal. Furthermore, the monomial expression given by our method is an equisingularity invariant of the ideal. As an outcome, we provide a geometric method to compute the integral closure of a planar ideal and we apply our algorithm to some families of complete ideals.

show abstract

Section: 1mentioning

confidence: 99%

Section: Multiplier Idealsmentioning

confidence: 99%

“…The logresolution of a can be computed using the algorithm from [2] and is represented by means of the following dual-graph: The divisor F such that a·O ′…”

Section: Multiplier Idealsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Monomial generators of complete planar ideals

2020

Self Cite

View full text Add to dashboard Cite

show abstract

Quadratic planar differential systems with algebraic limit cycles via quadratic plane Cremona maps

Alberich-Carramiñana

Ferragut

Llibre

2021

Advances in Mathematics

View full text Add to dashboard Cite

Bernstein-Sato polynomial of plane curves and Yano's conjecture

Blanco Fernández

View full text Add to dashboard Cite

The main aim of this thesis is the study of the Bernstein-Sato polynomial of plane curve singularities. In this context, we prove a conjecture posed by Yano about the generic b-exponents of a plane irreducible curve. In a part of the thesis, we study the Bernstein-Sato polynomial through the analytic continuation of the complex zeta function of a singularity. We obtain several results on the vanishing and non-vanishing of the residues of the complex zeta function. Using these results we obtain a proof of Yano's conjecture under the hypothesis that the eigenvalues of the monodromy are pair-wise different. In another part of the thesis, we study the periods of integrals in the Milnor fiber and their asymptotic expansion. These periods of integrals can be related to the b-exponents and can be constructed in terms of resolution of singularities. Using these techniques, we can present a proof for the general case of Yano's conjecture. In addition to the Bernstein-Sato polynomial, we also study the minimal Tjurina number of a plane irreducible curve and we answer in the positive a question raised by Dimca and Greuel on the quotient between the Milnor and Tjurina numbers. More precisely, we prove a formula for the minimal Tjurina number of a plane irreducible curve in terms of the multiplicities of the strict transform along its minimal resolution. From this formula, we obtain the positive answer to Dimca and Greuel question. This thesis also contains computational results for the theory of singularities on smooth complex surfaces. First, we describe an algorithm to compute log-resolutions of ideals on a smooth complex surface. Secondly, we provide an algorithm to compute generators for complete ideals on a smooth complex surface. These algorithms have several applications, for instance, in the computation of the multiplier ideals associated to an ideal on a smooth complex surface. El principal objectiu d'aquesta tesi és l'estudi del polinomi de Bernstein-Sato de singularitats de corbes planes. En aquest context, es demostra una conjectura proposada per Yano el 1982 sobre els \( b \)-exponents genèrics d'una corba plana irreductible. En una part d'aquesta tesi, s'estudia el polinomi de Bernstein-Sato utilitzant la continuació analítica de la funció zeta complexa d'una singularitat. S'obtenen diversos resultat sobre l'anul·lació i no anul·lació del residu de la funció zeta complexa d'una corba plana. Utilitzant aquests resultats, s'obté una demostració de la conjectura de Yano sota la hipòtesi de que els valors propis de la monodromia siguin diferents dos a dos. En un altre part de la tesi, s'estudien els períodes d'integrals en la fibra de Milnor i la seva expansió asimptòtica. Aquesta expansió asimptòtica dels períodes pot ser relacionada amb els b-exponents i pot ser construïda en termes de la resolució de singularitats. Utilitzant aquestes tècniques, es presenta una prova del cas general de la conjectura de Yano. A més a més del polinomi de Bernstein-Sato, també s'estudia el nombre de Tjurina mínim d'una corba plana irreductible i responem positivament a una pregunta formulada per Dimca i Greuel sobre el quocient entre els nombres de Milnor i Tjurina. Concretament, es demostra una fórmula pel nombre de Tjurina mínim en un classe d'equisingularitat de corbes planes irreductibles en termes de la seqüència de multiplicitats de la transformada estricta al llarg de la resolució minimal. A partir d'aquesta fórmula, s'obté la resposta positiva a la pregunta de Dimca i Greuel. Aquesta tesi també conté resultats computacionals per la teoria de singularitats en superfícies complexes llises. Primer, es descriu un algorisme que calcula la log-resolució d'ideals en un superfície complexa llisa. En segon lloc, es dona un algorisme per calcular generadors per ideals complets en una superfície complexa llisa. Aquests algorismes tenen diverses aplicacions, com per exemple, en el càlcul d'ideals multiplicadors associats a un ideal en una superfície complexa llisa.

show abstract

Effective computation of base points of ideals in two-dimensional local rings

Cited by 4 publications

References 10 publications

Monomial generators of complete planar ideals

Monomial generators of complete planar ideals

Quadratic planar differential systems with algebraic limit cycles via quadratic plane Cremona maps

Bernstein-Sato polynomial of plane curves and Yano's conjecture

Contact Info

Product

Resources

About