Unfolding Polynomial Maps at Infinity

Let f : Cn → C be a polynomial map. The polynomial describes a family of complex affine hypersurfaces f−1(c), c ∈ C. The family is locally trivial, so the hypersurfaces have constant topology, except at finitely many irregular fibers f−1(c) whose topology may differ from the generic or regular fiber of f . We would like to give a full description of the topology of this family in terms of easily computable data. This paper describes some progress. We will restrict mostly to the case that f has only isolated singularities. We show that the data then needed are local monodromy maps obtained by transporting a fixed generic fiber F around the irregular fibers, and the “Milnor fibers” of the singular points and singularities at infinity of f : these are certain submanifolds of F that describe the loss of topology at irregular fibers. It is convenient to subdivide this necessary data as follows. For each irregular fiber we need • the Milnor fibers associated with it; • the local monodromy for the irregular fiber restricted to each Milnor fiber; • the embeddings of the Milnor fibers into a fixed “reference” regular fiber F . The first two items are local ingredients, while the third is global. We are able to give complete computation of the local ingredients for n = 2 (see Theorem 5.1). The remaining problem for n = 2 is therefore the third item, although we obtain enough constraints that the complete topology can be sometimes be deduced. We use the Briançon polynomial as an illustrative example. In this case our general results quickly yield the previous homological monodromy computations of Artal-Bartolo, Cassou-Nogues, and Dimca [1] and Dimca and Nemethi [6]. Since our computations are geometric, we obtain sharper information (action on the intersection form, etc.). In fact, our computations strongly suggest a candidate description for the complete topology. However, this description remains conjectural, so the example illustrates both the strengths and current limitations of our approach.