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.KeywordsPolynomial EquationMinimal PolynomialSymmetric Bilinear FormMain LoopQuotient RingThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.