On the b-exponents of generic isolated plane curve singularities

E. Artal Bartolo, Pi. Cassou-Noguès, I. Luengo, and A. Melle-Hernández

Journal of Singularities
volume 18 (2018), 36-49

Received: 31 December 2017. Accepted: 22 February 2018.

DOI: 10.5427/jsing.2018.18d

Add a reference to this article to your citeulike library.

Abstract:

In 1982, Tamaki Yano proposed a conjecture predicting how is the set of b-exponents of an irreducible plane curve singularity germ which is generic in its equisingularity class. In 1986, Pi. Cassou-Noguès proved the conjecture for the one Puiseux pair case. The authors proved the conjecture for two Puiseux pairs germs whose complex algebraic monodromy has distinct eigenvalues. A natural problem induced by Yano's conjecture is, for a generic equisingular deformation of an isolated plane curve singularity germ to study how the set of b-exponents depends on the topology of the singularity. The natural generalization suggested by Yano's approach holds in suitable examples (for the case of isolated singularites which are Newton non-degenerated, commode and whose set of spectral numbers are all distincts). Morevover we show with an example that this natural generalization is not correct. We restrict to germs whose complex algebraic monodromy has distinct eigenvalues such that the embedded resolution graph has vertices of valency at most 3 and we discuss some examples with multiple eigenvalues.

Keywords:

Bernstein-Sato polynomial, b-exponents, Brieskorn lattice, improper integrals

Mathematical Subject Classification:

Primary 14F10, 32S40; Secondary 32S05, 32A30

Author(s) information:

E. Artal Bartolo
Departamento de Matemáticas-IUMA
Universidad de Zaragoza
c/ Pedro Cerbuna 12
50009 Zaragoza, SPAIN
email: artal@unizar.es

Pi. Cassou-Noguès
Institut de Mathématiques de Bordeaux
Université de Bordeaux
350, Cours de la Libération
33405, Talence Cedex 05, FRANCE
email: Pierrette.Cassou-nogues@math.u-bordeaux.fr

I. Luengo
ICMAT (CSIC-UAM-UC3M-UCM)
Dpto. de Álgebra, Geometría y Topología
Universidad Complutense de Madrid
Plaza de las Ciencias s/n, Ciudad Universitaria
28040 Madrid, SPAIN
email: iluengo@ucm.es

A. Melle-Hernández
Instituto de Matemática Interdisciplinar (IMI)
Dpto. de Álgebra, Geometría y Topología
Universidad Complutense de Madrid
Plaza de las Ciencias s/n, Ciudad Universitaria
28040 Madrid, SPAIN
email: amelle@ucm.es