Polar invariants of plane curve singularities: intersection theoretical approach

Gwoździewicz, Janusz; Lenarcik, Andrzej; Płoski, Arkadiusz

doi:10.1515/dema-2013-0246

Cited by 3 publications

(1 citation statement)

References 10 publications

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“…The first author of this note applied the Newton algorithm in Cano's version to determine the so-called polar quotients with their multiplicities [19]. A survey of results concerning polar invariants (quotients) is given in [12]. The more general are the so-called jacobian quotients [17].…”

Section: The Newton Algorithmmentioning

confidence: 99%

Effective proof of Guseĭn-Zade theorem that branches may be deformed with jump one

Lenarcik¹,

Masternak²

2022

Analitic and Algebraic Geometry 4

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Let f ∈ C{X, Y } be a reduced series which defines a singular branch f = 0 in a neighbourhood of zero in C 2 . Let h ≥ 1 be the number of characteristic exponets of a Puiseux root y(X) ∈ C{X} * of the equation f = 0. For any k ∈ {1, . . . , h} we define the series f k ∈ C{X, Y } generated by all terms of the series y(X) with orders strictly smaller than the k-th characteristic exponent. We consider a deformation Ft = f + tX ω 0 f ω 1 1 . . . f ω h h (t ̸ = 0, small) where ω 0 , ω 1 , . . . , ω h are nonnegative integers. Using a version of the Newton algorithm proposed by Cano we show how to choose exponents ω 0 , ω 1 , . . . , ω h to obtain the Milnor number of the deformation Ft smaller by one than the Milnor number of the branch f . We prove a version of Kouchnirenko theorem which is useful in computation the Milnor number.

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Section: The Newton Algorithmmentioning

confidence: 99%

Effective proof of Guseĭn-Zade theorem that branches may be deformed with jump one

Lenarcik¹,

Masternak²

2022

Analitic and Algebraic Geometry 4

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Topological invariants of plane curve singularities: Polar quotients and Łojasiewicz gradient exponents

2019

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In this paper, we study polar quotients and Lojasiewicz exponents of plane curve singularities, which are not necessarily reduced. We first show that the polar quotients is a topological invariant. We next prove that the Lojasiewicz gradient exponent can be computed in terms of the polar quotients, and so it is also a topological invariant. As an application, we give effective estimates of the Lojasiewicz exponents in the gradient and classical inequalities of polynomials in two (real or complex) variables.

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A discriminant criterion of irreducibility

Barroso

Gwoździewicz²

2012

Kodai Math. J.

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Polar invariants of plane curve singularities: intersection theoretical approach

Abstract: This article, based on the talk given by one of the authors at the Pierrettefest in Castro Urdiales in June 2008, is an overview of a number of recent results on the polar invariants of plane curve singularities.

Cited by 3 publications

References 10 publications

Effective proof of Guseĭn-Zade theorem that branches may be deformed with jump one

Effective proof of Guseĭn-Zade theorem that branches may be deformed with jump one

Topological invariants of plane curve singularities: Polar quotients and Łojasiewicz gradient exponents

A discriminant criterion of irreducibility

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