The Index of grad f(x,y)
نویسنده
چکیده
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 jij d ? 1. In this paper we show that i maxf1; d ? 3g. We also show that if all the level sets of f are compact, then i = 1, and otherwise jij 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 innnity. The index i is broken up into a sum of components i p;c corresponding to points p in the real line at innnity and limiting values c 2 Rf1g of the polynomial. The numbers i p;c are computed in three ways: geometrically, from a resolution of f(x; y), and from a Morsiication 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.
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