On the Seifert form at infinity associated with polynomial maps

Abstract

If apolynomial map $f$ : $C^n\to C$ has anice 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 $\Gamma \left(f\right)$ associated with $f$. In this paper we prove a Sebastiani-Thom type formula. Namely, if $f$ : $C^n\to C$ and $g:c^m\to C$ are "good" polynomials, and we define $h=f$$\oplus$$g$ : $C^{n+m}\to C$ by $h(x,y)=f\left(x\right)+g\left(y\right)$, then $\Gamma \left(h\right)=(-I{)}^{mn}\Gamma (f)$$\otimes \Gamma \left(g\right)$. This is the global analogue of the local result, proved independently by K. Sakamoto and P. Deligne for isolated hypersurface singularities.