Relative cohomology of polynomial mappings

Let F be a polynomial mapping from C to C with n > q. We study the De Rham cohomology of its fibres and its relative cohomology groups, by introducing a special fibre F−1(∞) ”at infinity” and its cohomology. Let us fix a weighted homogeneous degree on C[x1, ..., xn] with strictly positive weights. The fibre at infinity is the zero set of the leading terms of the coordinate functions of F . We introduce the cohomology groups Hk(F−1(∞)) of F at infinity. These groups enable us to compute all the other cohomology groups of F . For instance, if the fibre at infinity has an isolated singularity at the origin, we prove that every weighted homogeneous basis of Hn−q(F−1(∞)) is a basis of all the groups Hn−q(F−1(y)) and also a basis of the (n−q) relative cohomology group of F . Moreover the dimension of Hn−q(F−1(∞)) is given by a global Milnor number of F , which only depends on the leading terms of the coordinate functions of F .