On the shape of possible counterexamples to the Jacobian Conjecture

We improve the algebraic methods of Abhyankar for the Jacobian Conjecture in dimension two and describe the shape of possible counterexamples. We give an elementary proof of the result of Heitmann in [5], which states that gcd⁡(deg⁡(P),deg⁡(Q))≥16gcd⁡(deg⁡(P),deg⁡(Q))≥16 for any counterexample (P,Q)(P,Q). We also prove that gcd⁡(deg⁡(P),deg⁡(Q))≠2pgcd⁡(deg⁡(P),deg⁡(Q))≠2p for any prime p.