1995/06/01 by Alan H. Durfee, Durfee, Alan H.
Computer Science · Mathematics · #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Combinatorics #Degree (music) #Discrete mathematics #FOS: Mathematics #Homogeneous #Infinity #Limiting #Mathematical Dynamics and Fractals #Mathematical analysis #Mathematics #Physics #Polynomial #Polynomial and algebraic computation #Real line #Resolution (logic) #Type (biology) #Upper and lower bounds #alg-geom #math.AG
paper · pdf · doi:10.48550/arxiv.alg-geom/9506002
A thoroughly revised and hopefully more readable version; the main results are the same. 35 pages with 7 figures
openalex publication_date 1995/06/01 · arxiv created 1997/09/16 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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 ∈ \realinf 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.