Les singularités à l'infini des polynômes et les compactifications toriques

Abstract

We study the toric compactifications of fibers of a polynomial mapping in several complex variables and analyse their singularities which can appear at infinity. We compare severals possible definitions of such singularities. Essentially, these definitions are related to the topological triviality, the non-characteristic condition, the gradient condition and the absence of vanishing cycles at infinity. We generalize to the toric compactification set-up the results known for the projective compactification.