Fibre de Milnor motivique à l’infini et composition avec un polynôme non dégénéré  [ Motivic Milnor fiber at infinity and composition with a non-degenerate polynomial ]
Annales de l'Institut Fourier, Volume 62 (2012) no. 5, p. 1943-1981

Let k be a field of characteristic zero and P be a Laurent polynomial in d variables, with coefficients in k and non degenerate for its Newton polyhedron at infinity. Let (f l ) be d non constant functions with separated variables and defined on smooth varieties. As Guibert, Loeser and Merle in the local case, we compute in this article the motivic Milnor fiber at infinity of P(f) in terms of the Newton polyhedron at infinity of P. For P equal to the sum x 1 +x 2 , we obtained a Thom-Sebastiani formula. Then we can introduce a notion of motivic vanishing cycles of a function g for the infinite value denoted by S g,U Φ , and which verified, as in the local case, a convolution formula. In particular if g is the polynomial x 1 +...+x n +1/x 1 ...x n , we show that the spectrum S g,,U Φ is 1+t+..+t n which coincides with the spectrum at infinity of g considered by Douai and Sabbah.

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 du type Thom-Sébastiani. Ceci permet d’introduire une notion de cycles évanescents motiviques d’une fonction g pour la valeur infini notée S g,,U Φ , qui vérifie comme dans le cas local une formule de convolution. En particulier si g est le polynôme x 1 +...+x n +1/x 1 ...x n , nous montrons que le spectre de S g,,U Φ vaut 1+t+..+t n ce qui coïncide avec le spectre à l’infini de g considéré par Douai et Sabbah.

DOI : https://doi.org/10.5802/aif.2739
Classification:  14-99,  14J17,  14B05
Keywords: Algebraic Geometry, Singularities at infinity, Milnor fiber, motivic Milnor fiber, Thom-Sébastiani, nearby cycles.
@article{AIF_2012__62_5_1943_0,
     author = {Raibaut, Michel},
     title = {Fibre de Milnor motivique \`a l'infini et composition avec un polyn\^ome non d\'eg\'en\'er\'e},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {62},
     number = {5},
     year = {2012},
     pages = {1943-1981},
     doi = {10.5802/aif.2739},
     mrnumber = {3025157},
     zbl = {1266.14008},
     language = {fr},
     url = {http://www.numdam.org/item/AIF_2012__62_5_1943_0}
}
Raibaut, Michel. Fibre de Milnor motivique à l’infini et composition avec un polynôme non dégénéré. Annales de l'Institut Fourier, Volume 62 (2012) no. 5, pp. 1943-1981. doi : 10.5802/aif.2739. http://www.numdam.org/item/AIF_2012__62_5_1943_0/

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