In this chapter we first give a detailed description of the geometry of the support of polynomials in K[x, y] and the relation with the plane Jacobian conjecture. Then we introduce our main tools: directions and leading terms ℓρ,σ(P) corresponding to these directions (see Definition 4.2), the transformation of the shape of the support by certain automorphisms of K[x, y], and the very important element F.

Summary Chapter I

I.1 Polynomials in two variables and their support

I.2 The Jacobian determinant and its support

I.3 Plane Jacobian conjecture and small examples

I.4 Directions, leading terms, starting points and end points

I.5 The element F

I.6 Order on directions and shaping the support

I.7 The shape of counterexamples and (m,n)-pairs

I.8 The elements R,F

Table of R,F satisfying Theorem I.7.11, with deg(R) ≤ 16.