Celebratio Mathematica — Williams — Complete Bibliography

Complete Bibliography

Works connected to David C. Tischler

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M. Shub, D. Tischler, and R. F. Williams: “The Newtonian graph of a complex polynomial,” SIAM J. Math. Anal. 19 : 1 (1988), pp. 246–256. MR 924558 Zbl 0653.58013 article

In a recent paper [4] Smale posed as an important problem in complexity theory, characterization of the graph \( G_f \) of the Newtonian vector field \( N_f \) 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. In addition we give an example and prove a proposition to show this is necessary.

We present the proof as Theorem C in §5 using the topological characterization of analytic maps given by Stöilow [5] in 1929. 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.

David C. Tischler	Related
Michael Ira Shub	Related