The Newtonian Graph of a Complex Polynomial

In a recent paper [4] Smale posed as an important problem in complexity theory, characterization of the graph Gf of the Newtonian vector field Nf for a complex polynomial f. Such graphs are known to be connected and acyclic and Smale conjectured that these two properties completely characterize them. The purpose of this paper is to prove this conjecture, after adding an additional hypothesis (part 3 of the definition of “dynamic graph,” § 2). In additon we give an example and prove a proposition to show this is necessary (see “Counterexamples” in § 2). We present the proof as Theorem C in § 5 using the topological characterization of analytic maps given by Stöilow [5] in 1929. Bill Thurston pointed us in this direction, though considering the fact that G. T. Whyburn was the major professor of one of us, we should not have needed this help. In addition we present direct proofs of three special cases as Theorem A, Theorem B and Example 7. While this was being written an independent proof was given in the generic case (Theorem A) in [2]. Sections 1 and 2 are devoted to basic properties and to a list of examples designed to acquaint the reader (and the writers) with various aspects of Newtonian Graphs.