The Newtonian graph of a complex polynomial. (English) Zbl 0653.58013

Shub, Michael; Tischler, David; Williams, Robert F.

SIAM J. Math. Anal. 19, No. 1, 246-256 (1988).

The paper deals with Smale’s conjecture which concerns the characterization of the graph \(G_ f\) of the Newtonian vector field \(N_ f\) for a complex polynomial f, where \(G_ f\) is the graph whose vertices are zeros of f and f’ and whose directed edges are unstable manifolds of zeros of f’ which are not zeros of f. The authors define dynamic graphs which means a finite directed graph with two types of vertices: saddles and sinks, so that: at a sink, all edges are directed toward the sink; saddles have at least two outwardly directed edges; at a saddle, any two adjacent outwardly directed edges have at most one inwardly directed edge between them (called a saddle connection); each sink has a weight which is a positive integer (corresponding to the multiplicity of a zero of f). The main result says that any acyclic dynamic graph G with all saddles hyperbolic, no saddle connections and all weights \(=1\) is isotopic to some \(G_ f\) with f being a complex polynomial.

Reviewer: Yu.Kifer

MSC:

37D15	Morse-Smale systems
65D15	Algorithms for approximation of functions
37D99	Dynamical systems with hyperbolic behavior

Keywords:

Newtonian vector field; complex polynomial; saddle connections