On the Seifert form at infinity associated with polynomial maps

Abstract

If a polynomial map f:Cⁿ→ C has a nice behaviour at infinity (e.g. it is a“good polynomial”), then the Milnor fibration at infinity exists, in particular, one can define the Seifert form at infinity Γ(f) associated with f. In this paper we prove a Sebastiani-Thom type formula. Namely, if f:Cⁿ→ C and g:C^m→ C are“good”polynomials, and we define h=f+g:C^n+m→ C by h(x, y)=f(x)+g(y), then Γ(h)=(-1)^mnΓ(f) Γ(g). This is the global analogue of the local result, proved independently by K. Sakamoto and P. Deligne for isolated hypersurface singularities.