IJPAM: Volume 110, No. 4 (2016)

Title

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

Authors

Masukazu Hirata
Umezono 2-34-24, Tsukuba
Ibaraki 305-0045, JAPAN

Abstract

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 $\vert x\!+\!yi\vert$ 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 $n$th order equation obviously has $n$ roots.

History

Received: September 18, 2016
Revised: November 5, 2016
Published: November 9, 2016

AMS Classification, Key Words

AMS Subject Classification: 32Q55, 30C10, 65E05
Key Words and Phrases: integer-order unary polynomial, topological classification, special 3-dimensional coordinate system, real curve, complex curve

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Bibliography

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How to Cite?

DOI: 10.12732/ijpam.v110i4.9 How to cite this paper?

Source:
International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2016
Volume: 110
Issue: 4
Pages: 665 - 677


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