Preliminaries

Abstract

Polynomial maps have of course been common tools in algebraic geometry for many years. But more recently, motivated by the study of some tempting conjectures, the subject started to become a subbranch of affine algebraic geometry. Surprising new results were obtained, several conjectures were made and proofs of existing results were simplified etc. In this chapter we start developing some of the main tools which play a crucial role throughout this book. The first of these is the formal inverse function theorem, the algebraic analogue of the local inverse function theorem of calculus. For example we use it to describe the so-called Lefschetz-principle and the clearing map; both are techniques to reduce the study of polynomial maps over commutative ℚ-algebras to polynomial maps over such special rings as ℤ or ℂ.