Chapter 5 - Polynomial

Abstract

Polynomials arise naturally in many areas of mathematics and science. The solution of systems of polynomial equations, of course with multiple variables, demands efficient methods that clearly determine the solutions of such systems. For this purpose, there are mathematical procedures, among which numerical and algebraic methods stand out in engineering. Numerical methods allow finding approximate solutions while algebraic methods seek to reduce the systems of polynomial equations into a single polynomial equation in one unknown, which is simply an overwhelming task in which it is necessary to manipulate authentic labyrinths of equations. This chapter summarizes two numerical methods that allow one to determine the solutions of systems of polynomial equations: the Newton-Raphson method and the Newton-homotopy method. Moreover, the chapter includes methods based on algebraic geometry as well as the Sylvester dialytic method of elimination. The latter methods are appropriate when the systems of polynomial equations are composed of monomials and therefore one has the possibility to obtain symbolic solutions which are highly appreciated in the position analysis of mechanisms and robotic manipulators. Finally, other recommended strategies are the use of graphical methods, where possible, as well as the use of software such as Maple™ and Mathematica.™ In that sense, the availability of freely available specialized software such as Bertini [1] and PHCpack [2] represents potential computational tools that are highly advisable when the systems to be solved have solutions.