Summary Chapter I

The main result of this chapter is that if the Jacobian conjecture is false, then B ≥ 9, where

B := min{gcd(deg(P), deg(Q))},

and (P, Q) runs on the counterexamples. This result was already proven by Nagata in [11].

In order to prove B ≥ 9, we consider a minimal counterexample (P, Q), i. e., a pair of polynomials P,Q ∈ K[x, y], such that B = gcd(deg(P), deg(Q)). We transform the minimal counterexample into a standard (m, n)-pair in K[x, y] (see Definition 7.1). For the transformed pair there exist (ρ, σ)-homogeneous elements (see Definition 4.4) R, F satisfying very strong conditions (see Theorem 7.11). The support of the smallest R, F satisfying these conditions is shown in the following figure,

and deg(R) = 9 implies B ≥ 9.

The key point is the transformation of the minimal counterexample into a standard (m, n)-pair, which needs the most amount of work. This transformation is based on automorphisms of the algebra K[x, y], which transform the support of the polynomials, thus it transforms subsets of N0 × N0, using basic geometry on the plane. The shaping of the support depends on the multiplicity of the linear factors in the polynomials ℓρ,σ(P).

These multiplicities are strongly conditioned by the multiplicity of these linear factors in the (ρ, σ)-homogeneous polynomial F (see Proposition 5.5). This underlines the importance of the element F, which is the crucial new tool of our whole approach.

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