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Fibre de Milnor motivique à l’infini et composition avec un polynôme non dégénéré

Michel Raibaut (2012)

Annales de l’institut Fourier

Soit k un corps de caractéristique nulle, P un polynôme de Laurent en d variables, à coefficients dans k et non dégénéré pour son polyèdre de Newton à l’infini. Soit d fonctions non constantes f l à variables séparées et définies sur des variétés lisses. A la manière de Guibert, Loeser et Merle, dans le cas local, nous calculons dans cet article, la fibre de Milnor motivique à l’infini de la composée P ( f ) en termes du polyèdre de Newton à l’infini de P . Pour P égal à la somme x 1 + x 2 nous obtenons une formule...

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...

On the Łojasiewicz Exponent near the Fibre of a Polynomial

Grzegorz Skalski (2004)

Bulletin of the Polish Academy of Sciences. Mathematics

The equivalence of the definitions of the Łojasiewicz exponent introduced by Ha and by Chądzyński and Krasiński is proved. Moreover we show that if the above exponents are less than -1 then they are attained at a curve meromorphic at infinity.

Relative exactness modulo a polynomial map and algebraic ( p , + ) -actions

Philippe Bonnet (2003)

Bulletin de la Société Mathématique de France

Let F = ( f 1 , ... , f q ) be a polynomial dominating map from n to  q . We study the quotient 𝒯 1 ( F ) of polynomial 1-forms that are exact along the generic fibres of F , by 1-forms of type d R + a i d f i , where R , a 1 , ... , a q are polynomials. We prove that 𝒯 1 ( F ) is always a torsion [ t 1 , ... , t q ] -module. Then we determine under which conditions on F we have 𝒯 1 ( F ) = 0 . As an application, we study the behaviour of a class of algebraic ( p , + ) -actions on n , and determine in particular when these actions are trivial.

The Łojasiewicz gradient inequality in a neighbourhood of the fibre

Janusz Gwoździewicz, Stanisław Spodzieja (2005)

Annales Polonici Mathematici

Some estimates of the Łojasiewicz gradient exponent at infinity near any fibre of a polynomial in two variables are given. An important point in the proofs is a new Charzyński-Kozłowski-Smale estimate of critical values of a polynomial in one variable.

[unknown]

Matthias Leuenberger (0)

Annales de l’institut Fourier

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