Solving Polynomial Equations

Abstract

In this chapter we will discuss several approaches to solving systems of polynomial equations. First, we will discuss a straightforward attack based on the elimination properties of lexicographic Gröbner bases. Combining elimination with numerical root-finding for one-variable polynomials we get a conceptually simple method that generalizes the usual techniques used to solve systems of linear equations. However, there are potentially severe difficulties when this approach is implemented on a computer using finite-precision arithmetic. To circumvent these problems, we will develop some additional algebraic tools for root-finding based on the algebraic structure of the quotient rings k[x ₁,..., x _n ]/I. Using these tools, we will present alternative numerical methods for approximating solutions of polynomial systems and consider methods for real root-counting and root-isolation. In Chapters 3, 4 and 7, we will also discuss polynomial equation solving. Specifically, Chapter 3 will use resultants to solve polynomial equations, and Chapter 4 will show how to assign a well-behaved multiplicity to each solution of a system. Chapter 7 will consider other numerical techniques (homotopy continuation methods) based on bounds for the total number of solutions of a system, counting multiplicities.