The index of $grad f(x,y)$
| dc.creator | Durfee, Alan H. | |
| dc.date | 1995-06-01 | |
| dc.date | 1997-09-16 | |
| dc.date.accessioned | 2026-07-07T08:57:58Z | |
| dc.date.available | 2026-07-07T08:57:58Z | |
| dc.description | 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 $c$. | |
| dc.description | A thoroughly revised and hopefully more readable version; the main results are the same. 35 pages with 7 figures | |
| dc.identifier | https://arxiv.org/abs/alg-geom/9506002 | |
| dc.identifier | http://arxiv.org/abs/alg-geom/9506002 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/147140 | |
| dc.subject | Algebraic Geometry | |
| dc.title | The index of $grad f(x,y)$ | |
| dc.type | text |