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

Abstract

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.