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

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