In this chapter, we present an important algorithmic approach to dealing with polynomials in several variables. Hironaka (1964) introduced in his work on resolution of singularities over ℂ—for which he received the Fields medal, the “Nobel prize” in mathematics—a special type of basis for polynomial ideals, called “standard basis”. Bruno Buchberger (1965) invented them independently in his Ph. D. thesis, and named them Gröbner bases after his thesis advisor Wolfgang Gröbner. They are a vital tool in modern computational algebraic geometry.

We start with two examples, one from robotics and one illustrating “automatic” proofs of theorems in geometry. We then introduce the basic notions of orders on monomials and the resulting division algorithm. Next come two important theorems, by Dickson and by Hilbert, that guarantee finite bases for certain ideals. Then we can define Gröbner bases and Buchberger's algorithm to compute them.

The end of this chapter presents two “geometric” applications: implicitization of algebraic varieties and solution of systems of polynomial equations. While these fall naturally into the realm of manipulating polynomials, the examples in Sections 24.1 and 24.2 below are less expected: logical proof systems and analysis of parallel processes. We cannot even mention numerous other applications, for example, in tiling problems and term rewriting. We finish with some facts—without proof—on the cost of computing Gröbner bases.