The index of $grad f(x,y)$

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 $|i| \leq d-1$. In this paper we show that $ i \leq max \{1, d-3 \}$. We also show that if all the level sets of $f$ are compact, then $i = 1$, and otherwise $ |i| \leq \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 $i_{p,c}$ corresponding to points $p$ in the real line at infinity and limiting values $c \in \realinf$ 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 $c$.