F eb 2 00 5 NEARBY CYCLES AND COMPOSITION WITH A NON - DEGENERATE POLYNOMIAL

Let Xj be smooth varieties over a field k of characteristic zero, for 1 ≤ j ≤ p. Consider a family f of p functions fj : Xj → A 1 k. We shall denote also by fj the function on the product X = ∏ j Xj obtained by composition with the projection. We denote by X0(f) the set of common zeroes in X of the functions fj. Let P ∈ k[y1, . . . , yp] be a polynomial, which we assume to be non-degenerate with respect to its Newton polyhedron. In the present note we shall compute the motivic nearby cycles SP (f) of the composed function P (f) on X0(f) as a sum over the set of compact faces δ of the Newton polyedron of P . For every such δ, let us denote by Pδ the corresponding quasi-homogeneous polynomial. We associate to such a quasi-homogeneous polynomial a convolution operator ΨPδ , which in the special case where Pδ is the polynomial Σ = y1 + y2 is nothing but the operator ΨΣ considered in [9]. For such a compact face δ, one may also define generalized nearby cycles S σ(δ)