On Sectional Newtonian Graphs
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ON SECTIONAL NEWTONIAN GRAPHS Zening Fan, Chengdu, Suo Zhao, Shanghai Received February 7, 2020. Published online August 17, 2020.
Abstract. In this paper, we introduce the so-called sectional Newtonian graphs for univariate complex polynomials, and study some properties of those graphs. In particular, we list all possible sectional Newtonian graphs when the degrees of the polynomials are less than five, and also show that every stable gradient graph can be realized as a polynomial sectional Newtonian graph. Keywords: sectional Newtonian graph; level set; partition MSC 2020 : 05C75, 53C43
1. Introduction In 1985, Smale in [8] introduced the Newtonian graph Nf for any complex univariate polynomial f ∈ C[z] as follows. Vertices of Nf are f −1 (0) ∪ fz−1 (0), which consist of zeros of f and its holomorphic derivative fz := df /dz. Edges of Nf are the degenerate curves of flow of the associated Newtonian vector field Vf (z) := −
f (z) , fz (z)
z ∈ C.
The word “Newtonian” comes from the fact that those vector fields are related to Newton’s method, which was discussed in Newton’s own book Method of Fluxions. A Newtonian graph Nf is completely determined by the polynomial f . However, the answer to the converse problem is not obvious, i.e.: Given a graph G, is it the Newtonian graph Nf of some f ∈ C[z]? Based on the contribution of Smale (see [8]), various successors went deeper into Newtonian graphs and provided their partial answers. In 1988, Shub, Tischler and The authors were supported by NSFC 11801385, 11501530, 11501383 and 11571242 during the preparation of this paper. DOI: 10.21136/CMJ.2020.0049-20
605
William in [7] proved that given an acyclic dynamic graph G ⊂ R2 , there always exists a polynomial f such that the Newtonian graph Nf is isotopic to G, where the dynamic graph is a finite directed graph with two types of vertices, which we call saddles and sinks subject to some conditions (see [7], page 251). In [5], Kahn detailed two families of Newtonian graphs of complex polynomials. Besides, Jongen, Jonker and Twilt in [4] proved some necessary and sufficient conditions to judge whether a plane graph is equivalent to a Newtonian graph. In 1995, Stefánsson generalized the definition of Newtonian graphs in his Ph.D. dissertation (see [9] or [6]) in two directions: on the one hand, he defined those graphs for rational functions on C; on the other hand, he also defined those graphs on general Riemann surfaces which are not necessarily closed. Although the Newtonian graph Nf reflects many important properties of f , deficiencies still exist. For example, as mentioned in [7], multiple zeros of f have no geometric effect on the corresponding sinks, instead, only the velocity of the flow increases. The main purpose of this paper is to introduce sectional Newtonian graphs, SNG for short, on the Riemann sphere. Roughly speaking, it is a combinatorical description of a complex polynomial (or meromorphic function) via some specified holomorphic pullback CW-structures. This paper is organized as follows. We devote Sec
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