A TOPOLOGICAL CLASSIFICATION OF INTEGER-ORDER UNARY POLYNOMIALS WITH REAL COEFFICIENTS BY NUMERICAL EXEMPLIFICATION

From the 2nd to 7th order of integer-order unary polynomials with real coefficients were topologically classified by numerical exemplification of curves in a special 3-dimensional coordinate system whose 3 axes were the real part x, the imaginary part yi and the real part of the polynomial Re{f(x+yi)} in the case that the imaginary part of the polynomial is 0. The polynomials of order n gave n curves (one real curve and n−1 complex curve(s)), and topological diversity of the curves existed in a small |x+yi| region and was characterized by n and several indices regarding both configuration and connection among the curves. A cross-section of the curves at any plane of f(x+yi)=real constant gave n points when their multiplicity was considered. This graphically showed that the nth order equation obviously has n roots. AMS Subject Classification: 32Q55, 30C10, 65E05


Introduction
The nth order of integer-order unary polynomials with real coefficients have n real and/or complex roots.This can be graphically shown by curves in a special 3-dimensional coordinate system whose 3 axes are the real part x, the imaginary part yi and the real part of the polynomial Re{f (x + yi)} in the case that the imaginary part of the polynomial is 0. This method has been independently discovered by several persons [1][2][3][4] and also by the author.This method is also expected to topologically classify the polynomials, although the classification has been completed only in the case of the 2nd and 3rd orders [3].In this paper, the author newly classifies the 4th to 7th order polynomials.
The nth order unary polynomial is limited to real numbers: Im{f (z)} = 0.In the region where |x+yi| is sufficiently large, the highest (nth) order term becomes dominant and the polynomial is approximated as f (z) ≈ a n z n .This cyclotomic equation gives the following simple n curves: where j = 1, 2, . . ., n.
Therefore, topological diversity does not exist in this region.On the contrary, topological diversity exists in the small |x+yi| region.
In this paper, the topological classification was carried out by numerical exemplification using a special calculating and drawing program.

Calculating and Drawing Program
The author prepared the special calculating and drawing computer program, which calculates a polynomial with given values of real coefficients and given drawing ranges of variables x, y and f (z) and draws n curves of the calculated values of the polynomials in the special 3-dimensional coordinate system and their projected n carves in the typical 2-dimensional complex coordinate system (x -yi plane).
The outline of the program is as follows: (1) Real curve (between x and f (z), y = 0).
In the drawing range of x, f (z) was calculated from z = x + 0i = x as an explicit function.
By sequential use of the real value c in the drawing range of f (z), f (z) = c was numerically solved as an implicit function, and the corresponding root x+yi was obtained.Here, f (z) of only the y ≥ 0 region in the ranges of x and y was solved because of the conjugacy of complex roots.Moreover, the continuity of values of roots and the similarity of tangential directions at near points on one curve drawn by roots about sequential c values were used in order to select the next root correctly also at the branch point of curves.

Searching of Diversity
Topological diversity of an nth order polynomial in the special 3-dimensional coordinate system is characterized by the mutual configuration and connection of n curves, as shown in the next sub-section.The author previously considered topological possibilities of the configuration and connection, showed them by numerical exemplification, and then classified them.Here, the exemplification of the a n > 0 case is sufficient and only the a n = 1 case was applied.
In order to easily obtain numerical examples of curves, coefficients a k of f (z) were determined by trial-and-error about n roots of f (z) = 0 (configuration of n roots in x -yi plane).In this determination, multiple roots and symmetry of curves were preferentially used, if they could be used.

Obtained Diversity
Table 1 shows the obtained topological classification which was characterized by the configuration and connection among n curves (one real curve and n−1 complex curve(s)).In this classification, the following notation of curves, as in Figure 1 (here, Re{f (z)} in the case of Im{f (z)} = 0 was simply denoted by f (z)), and following related indices were used: C a : real curve, one curve only C b : complex curve(s), directly connect to real curve C a C c : complex curve(s), indirectly connect to real curve C a (connect to C b or other C c ) C d : complex curve(s), do not connect to real curve C a Figure 2 shows examples of curves classified in Table 1.Here, as n becomes larger, the value of |f (z)| becomes much larger in the large |x + yi| region; therefore, the value of f (z) was arbitrarily reduced in each example, and then drawing ranges were set as −8 ≤ x ≤ 8, −8 ≤ y ≤ 8 and −10 ≤ f (z) ≤ 10.
Table 2 shows each polynomial of each example in Figure 2 by a formula  Topological properties of the real curve and complex curve(s) shown in Figure 2 are summarized as follows: (2) When n ≥ 3, there is/are class(es) having independent complex curves that do not branch from the real curve, and moreover, when n is an odd number, there is/are class(es) having the independent real curve that does not branch to complex curves at all.(3) When n ≥ 4, there is/are class(es) having both branch point(s) among the real curve and complex curves and branch points among plural complex curves.  2 (4) When n ≥ 5, there is/are class(es) having branch points among plural complex curves that are independent on real curve.
(5) At a sufficiently large |x+yi| region that is far from the original point, upward curve(s) and downward curve(s) are alternately arranged (this corre- sponds to the asymptotic behavior of f (z) ≈ a n z n ).

Transition of Shape of Curves by the Difference in the Value of Coefficient
Generally, branching or not is caused by small differences in the value of coefficient, as shown in Figure 3.The left 3 sub-figures show transitions from branching (from two m 1 = 2 branches (3101.9) to one m 1 = 3 branch (3102)) of the real curve and two complex curves to independent them (3102.1) of n = 3, and the right 3 sub-figures show transitions from branching of the real curve and three complex curves (4201) to branching of the real curve and one complex curve and two independent complex curves (left side (4200.9)or right side (4201.1)) of n = 4.

Changing of Shape of Curves by Increasing of Order
The polynomial is described also as )z + a 0 , and this shows that increasing of the order n is a repeating of both As these examples, multiplying by z corresponds that one new complex curve approaches from an infinite distance and adds to the existing curves.Here, the previous adding of a k (a 1 = −1 (upper 4 sub-figures) or a 1 = +1 (lower 4 sub-figures) in Figure 4) is just a translation, but the value of a k brings a diversity of curves after the next multiplying by z.

Figure 1 :
Figure 1: Coordinate systems and notation of curves

Table 1 :
Topological classification of integer-order unary polynomials with real coefficients containing n roots, and each value of factor r of the reduction.In this table, examples containing root(s) with 3 digits after the decimal point show that these examples exist only in a very narrow range of values and the root(s) were

Table 2 :
Formulas of polynomials of examples in Figure