Visualization of graphs of complex functions and exact geometric method of finding complex roots of a polynomial

S. Trofimov, O. Trofimova
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Abstract

The paper describes an application that allows to visualize four-dimensional graphs of a complex variable function. Three coordinates of the graph are Cartesian, and fourth is the parametric coordinate. Graphs of complex polynomials are considered in detail. We demonstrates the well-known basic theorem of algebra about the number of polynomial roots. The study of graphs of complex polynomials made it possible to construct an exact geometric algorithm for finding the real and complex roots of a polynomial on the same plane. The algorithm assumes the construction of a graph of the main polynomial and graphs of two auxiliary functions. The application of this method is considered in detail for a cubic polynomial. In this case the method has exceptional features in comparison with polynomials of other degrees. Taking into account the well-known expressions for the roots of polynomials of order 3 and 4, the auxiliary graphs of the method have exact formulas for polynomials with order from 3 to 10.
复数函数图的可视化和精确的几何方法找到一个多项式的复根
本文描述了一个应用程序,允许可视化一个复杂的变量函数的四维图形。图的三个坐标是笛卡尔坐标,第四个是参数坐标。复多项式图的详细考虑。我们证明了代数中关于多项式根个数的著名基本定理。对复数多项式图的研究使得在同一平面上找到多项式的实根和复根的精确几何算法成为可能。该算法假定构造一个主多项式的图和两个辅助函数的图。详细讨论了该方法对三次多项式的应用。在这种情况下,与其他次数的多项式相比,该方法具有特殊的特征。考虑到众所周知的3阶和4阶多项式的根表达式,该方法的辅助图具有3到10阶多项式的精确公式。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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