ABSTRACT
The title’s subject is the algorithmic approach to algebra: arithmetic with numbers, polynomials, matrices, differential polynomials, such as y ′′ + (1/2 + x4/4)y, truncated series, and algebraic sets, i.e., quantified expressions such as ∃x ∈ R : x4 + p · x + q = 0, which describes a subset of the two-dimensional space with coordinates p and q for which the given quartic equation has a real root. Algorithms that manipulate such objects are the backbone of modern symbolic mathematics software such as the Maple and Mathematica systems, to name but two among many useful systems. This chapter restricts itself to algorithms in four areas: linear matrix algebra, root finding of univariate polynomials, solution of systems of nonlinear algebraic equations, and polynomial factorization.
