Global index of real polynomials

Abstract

We develop two methods for expressing the global index of the gradient of a 2 variable polynomial function $f$ : in terms of the atypical fibres of $f$ , and in terms of the clusters of Milnor arcs at infinity. These allow us to derive upper bounds for the global index, in particular refining the one that was found by Durfee in terms of the degree of $f$ .