Methods and Applications of Analysis

Volume 24 (2017)

Number 2

Special issue dedicated to Henry B. Laufer on the occasion of his 70th birthday: Part 2

Guest Editors: Stephen S.-T. Yau (Tsinghua University, China); Gert-Martin Greuel (University of Kaiserslautern, Germany); Jonathan Wahl (University of North Carolina, USA); Rong Du (East China Normal University, China); Yun Gao (Shanghai Jiao Tong University, China); and Huaiqing Zuo (Tsinghua University, China)

Bernstein polynomial of $2$-Puiseux pairs irreducible plane curve singularities

Pages: 185 – 214

DOI: https://dx.doi.org/10.4310/MAA.2017.v24.n2.a2

Authors

E. Artal Bartolo (Departamento de Matemáticas, IUMA, Universidad de Zaragoza, Zaragoza, Spain)

Pi. Cassou-Nogués (Institut de Mathématiques de Bordeaux, Université de Bordeaux, Talence, France)

I. Luengo (Departamento de Álgebra, Universidad Complutense, Ciudad Universitaria, Madrid, Spain)

A. Melle-Hernández (Instituto de Matemática Interdisciplinar (IMI), Departamento de Álgebra, Universidad Complutense, Ciudad Universitaria, Madrid, Spain)

Abstract

In 1982, Tamaki Yano proposed a conjecture predicting the set of $b$-exponents of an irreducible plane curve singularity germ which is generic in its equisingularity class. In 1986, the second author proved the conjecture for the one Puiseux pair case. In [1], we proved the conjecture for the case in which the germ has two Puiseux pairs and its algebraic monodromy has distinct eigenvalues. In this article we aim to study the Bernstein polynomial for any function with two Puiseux pairs and its algebraic monodromy has distinct eigenvalues. In particular the set of all common roots of their corresponding Bernstein polynomials is also explicitely given. We provide also bounds for some analytic invariants of singularities and illustrate the computations in suitable examples.

Keywords

Bernstein polynomial, $b$-exponents, improper integrals

2010 Mathematics Subject Classification

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

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E. Artal Bartolo was partially supported by the grant MTM2013-45710-C02-01-P and MTM2016-76868-C2-2-P.

Pi. Cassou-Nogués was partially supported by MTM2013-45710-C02-01-P, MTM2013-45710-C02-02-P, MTM2016-76868-C2-1-P and MTM2016-76868-C2-2-P.

I. Luengo was partially supported by the grant MTM2013-45710-C02-02-P and MTM2016-76868-C2-1-P.

A. Melle-Hernández was partially supported by the grant MTM2013-45710-C02-02-P and MTM2016-76868-C2-1-P.

Received 30 October 2016

Accepted 14 September 2017

Published 3 January 2018