The index of [formula omitted

Abstract

Let f(x, y) be a real polynomial of degree d with isolated critical points, and let i be the index of grad f around a large circle containing the critical points. An elementary argument shows that | i|⩽ d−1. In this paper we show that i⩽ max{1, d−3} . We also show that if all the level sets of f are compact, then i=1, and otherwise |i|⩽d R −1 where d R is the sum of the multiplicities of the real linear factors in the homogeneous term of highest degree in f. The technique of proof involves computing i from information at infinity. The index i is broken up into a sum of components i p, c corresponding to points p in the real line at infinity and limiting values c∈ R ∪{∞} of the polynomial. The numbers i p, c are computed in three ways: geometrically, from a resolution of f(x, y) , and from a Morsification of f(x, y) . The i p, c also provide a lower bound for the number of vanishing cycles of f(x, y) at the point p and value c.