Unfolding polynomial maps at infinity

Abstract

We give a topological model for a polynomial map from $\C^n$ to $\C$ in the neighborhood of a fiber with isolated singularities. This is motivated out of the ``unfolding of links'' described earlier by the first author and Lee Rudolph. The topological model gives a useful encoding of the local and global monodromy for the map. The new ingredients --- the topology of the singularities at infinity of an irregular fiber, as encoded by the Milnor fibers at infinity and the monodromy maps on these Milnor fibers --- are recoverable for n=2 in full from the link at infinity of the irregular fiber. They can thus be computed completely in terms of splice diagrams. We illustrate our results by computing the global monodromy of the Briancon polynomial.