Relative differential forms and complex polynomials

Bonnet, Philippe; Dimca, Alexandru

doi:10.1016/s0007-4497(00)01055-1

Cited by 19 publications

(12 citation statements)

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“…The fact the C( f ) is torsion free follows from Proposition 7. (ii) in [2]. Indeed, it is shown there that the localization morphism…”

Section: The Milnor Algebra and The Brieskorn Modulementioning

confidence: 93%

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On the Torsion of Brieskorn Modules of Homogeneous Polynomials

Shabbir¹

2008

Rend. Sem. Mat. Univ. Padova

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-Let f P C[X 1 ; :::; X n ] be a homogeneous polynomial and B( f ) be the corresponding Brieskorn module. We describe the torsion of the Brieskorn module B( f ) for n 2 and show that any torsion element has order 1. For n > 2, we find some examples in which the torsion order is strictly greater than 1. The Milnor algebra and the Brieskorn module.Let f P R C[x 1 ; :::; x n ] be a homogeneous polynomial of degree d > 1.Then the Koszul complex of the partial derivatives f j @f @x j ; j 1; :::; n in R can be identified to the complexwhere V j denotes the regular differential forms of degree j on C n . Let J f be the Jacobian ideal spanned by the partial derivatives f j , j 1; :::; n, in R and M( f ) R=J f be the Milnor algebra of f . One has the following obvious isomorphism of graded vector spacesHere, for any graded C[t]-module M, the shifted module M(m) is defined by setting M(m) s M ms for all s P Z:We define the (algebraic) Brieskorn module as the quotient

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“…The fact the C( f ) is torsion free follows from Proposition 7. (ii) in [2]. Indeed, it is shown there that the localization morphism…”

Section: The Milnor Algebra and The Brieskorn Modulementioning

confidence: 93%

“…Sometimes B( f ) is denoted by G (0) f and C( f ) by G (À1) f , see [2], [6]. One has the following basic relation between the Milnor algebra and the Brieskorn module, see [7], Prop.…”

Section: The Milnor Algebra and The Brieskorn Modulementioning

confidence: 99%

On the Torsion of Brieskorn Modules of Homogeneous Polynomials

Shabbir¹

2008

Rend. Sem. Mat. Univ. Padova

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“…Brieskorn in [Br] introduced three O ðC;0Þ -modules H; H 0 ; H 00 in the context of singularity of holomorphic functions f : ðC n ; 0Þ-ðC; 0Þ (The H used in this paper is the equivalent of Brieskorn's H 0 ). After him these modules are called the Brieskorn lattices and recently the similar notions in a global context are introduced by many authors (see [Sa,DS,BD,Mo3]). In the context of differential equations H appears in the works of Petrov [Pe] on deformation of Hamiltonian equations of the type dðy 2 þ pðxÞÞ; where p is a polynomial in x and is named by Gavrilov in [Ga] the Petrov module.…”

Section: Article In Pressmentioning

confidence: 99%

Center conditions: rigidity of logarithmic differential equations

Movasati

2004

Journal of Differential Equations

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In this paper, we prove that any degree d deformation of a generic logarithmic polynomial differential equation with a persistent center must be logarithmic again. This is a generalization of Ilyashenko's result on Hamiltonian differential equations. The main tools are Picard-Lefschetz theory of a polynomial with complex coefficients in two variables, specially the Gusein-Zade/A'Campo's theorem on calculating the Dynkin diagram of the polynomial, and the action of Gauss-Manin connection on the so-called Brieskorn lattice/ Petrov module of the polynomial. We will also generalize J.P. Francoise recursion formula and ðÃÞ condition for a polynomial which is a product of lines in a general position. Some applications on the cyclicity of cycles and the Bautin ideals will be given. r

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“…This paper is dedicated to the study of the tangential center problem on zerocycles: Problem 1. 3. Tangential center problem on zero-cycles.…”

Section: Introductionmentioning

confidence: 99%