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Classification analytique de structures de Poisson

Philipp Lohrmann (2009)

Bulletin de la Société Mathématique de France

Notre étude porte sur une catégorie de structures de Poisson singulières holomorphes au voisinage de 0 n et admettant une forme normale formelle polynomiale i.e. un nombre fini d’invariants formels. Les séries normalisantes sont divergentes en général. On montre l’existence de transformations normalisantes holomorphes sur des domaines sectoriels de la forme a < arg x R < b , où x R est un monôme associé au problème. Il suit une classification analytique.

Łojasiewicz exponent of the gradient near the fiber

Ha Huy Vui, Nguyen Hong Duc (2009)

Annales Polonici Mathematici

It is well-known that if r is a rational number from [-1,0), then there is no polynomial f in two complex variables and a fiber f - 1 ( t ) such that r is the Łojasiewicz exponent of grad(f) near the fiber f - 1 ( t ) . We show that this does not remain true if we consider polynomials in real variables. More exactly, we give examples showing that any rational number can be the Łojasiewicz exponent near the fiber of the gradient of some polynomial in real variables. The second main result of the paper is the formula...

Łojasiewicz exponents and singularities at infinity of polynomials in two complex variables

Janusz Gwoździewicz, Arkadiusz Płoski (2005)

Colloquium Mathematicae

For every polynomial F in two complex variables we define the Łojasiewicz exponents p , t ( F ) measuring the growth of the gradient ∇F on the branches centered at points p at infinity such that F approaches t along γ. We calculate the exponents p , t ( F ) in terms of the local invariants of singularities of the pencil of projective curves associated with F.

Multiplicity and the Łojasiewicz exponent

S. Spodzieja (2000)

Annales Polonici Mathematici

We give a formula for the multiplicity of a holomorphic mapping f : n Ω m , m > n, at an isolated zero, in terms of the degree of an analytic set at a point and the degree of a branched covering. We show that calculations of this multiplicity can be reduced to the case when m = n. We obtain an analogous result for the local Łojasiewicz exponent.

Puiseux series polynomial dynamics and iteration of complex cubic polynomials

Jan Kiwi (2006)

Annales de l’institut Fourier

We let 𝕃 be the completion of the field of formal Puiseux series and study polynomials with coefficients in 𝕃 as dynamical systems. We give a complete description of the dynamical and parameter space of cubic polynomials in 𝕃 [ ζ ] . We show that cubic polynomial dynamics over 𝕃 and are intimately related. More precisely, we establish that some elements of 𝕃 naturally correspond to the Fourier series of analytic almost periodic functions (in the sense of Bohr) which parametrize (near infinity) the quasiconformal...

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