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

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.