Abstract. 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 [7], which states that gcd(deg(P ), deg(Q)) ≥ 16 for any counterexample (P, Q). We also prove that gcd(deg(P ), deg(Q)) = 2p for any prime p and analyze thoroughly the case 16, adapting a reduction of degree technique introduced by Moh.
We prove that the Jacobian conjecture is false if and only if there exists a solution to a certain system of polynomial equations. We analyse the solution set of this system. In particular we prove that it is zero dimensional.2010 Mathematics Subject Classification. primary 14R15; secondary 13P15, 13F20.
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