Finding All Solutions to a System of Polynomial Equations
نویسندگان
چکیده
Given a polynomial equation of degree d over the complex domain, the Fundamental Theorem of Algebra tells us that there are d solutions, assuming that the solutions are counted by multiplicity. These solutions can be approximated by deforming a standard «th degree equation into the given equation, and following the solutions through the deformation. This is called the homotopy method. The Fundamental Theorem of Algebra can be proved by the same technique. In this paper we extend these results and methods to a system of n polynomial equations in n complex variables. We show that the number of solutions to such a system is the product of the degrees of the equations (assuming that infinite solutions are included and solutions are counted by multiplicity)*. The proof is based on a homotopy, or deformation, from a standard system of equations with the same degrees and known solutions. This homotopy provides a computational method of approximating all solutions. Computational results demonstrating the feasibility of this method are also presented.
منابع مشابه
Adomian Polynomial and Elzaki Transform Method of Solving Fifth Order Korteweg-De Vries Equation
Elzaki transform and Adomian polynomial is used to obtain the exact solutions of nonlinear fifth order Korteweg-de Vries (KdV) equations. In order to investigate the effectiveness of the method, three fifth order KdV equations were considered. Adomian polynomial is introduced as an essential tool to linearize all the nonlinear terms in any given equation because Elzaki transform cannot handle n...
متن کاملSolving Single Phase Fluid Flow Instability Equations Using Chebyshev Tau- QZ Polynomial
In this article the instability of single phase flow in a circular pipe from laminar to turbulence regime has been investigated. To this end, after finding boundary conditions and equation related to instability of flow in cylindrical coordination system, which is called eigenvalue Orr Sommerfeld equation, the solution method for these equation has been investigated. In this article Chebyshev p...
متن کاملA Numerical Approach for Solving of Two-Dimensional Linear Fredholm Integral Equations with Boubaker Polynomial Bases
In this paper, a new collocation method, which is based on Boubaker polynomials, is introduced for the approximate solutions of a class of two-dimensional linear Fredholm integral equationsof the second kind. The properties of two-dimensional Boubaker functions are presented. The fundamental matrices of integration with the collocation points are utilized to reduce the solution of the integral ...
متن کاملNumerical solution for the risk of transmission of some novel coronavirus (2019-nCov) models by the Newton-Taylor polynomial solutions
In this paper we consider two type of mathematical models for the novel coronavirus (2019-nCov), which are in the form of a nonlinear differential equations system. In the first model the contact rate, , and transition rate of symptomatic infected indeviduals to the quarantined infected class, , are constant. And in the second model these quantities are time dependent. These models are the...
متن کاملNumerical solution of a system of fuzzy polynomial equations by modified Adomian decomposition method
In this paper, we present some efficient numerical algorithm for solving system of fuzzy polynomial equations based on Newton's method. The modified Adomian decomposition method is applied to construct the numerical algorithms. Some numerical illustrations are given to show the efficiency of algorithms.
متن کامل