The index of grad f(x, y)

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|⩽dR−1 where dR 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 ip,c corresponding to points p in the real line at infinity and limiting values c∈R∪{∞} of the polynomial. The numbers ip,c are computed in three ways: geometrically, from a resolution of f(x, y), and from a Morsification of f(x, y). The ip,c also provide a lower bound for the number of vanishing cycles of f(x, y) at the point p and value c.