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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We define a type of crossed product over braided Hopf algebras, which generalizes the ones introduced by Blattner, Cohen and Montgomery and Doi and Takeuchi, and we study some of its properties. For instance, we prove a Maschke's Theorem for these new crossed products and under suitable hypothesis we construct a natural Morita context which extends the one obtained by Cohen, Fischman and Montgomery.

We find three families of twisting maps of K m with K n. One of them is related to truncated quiver algebras, the second one consists of deformations of the first and the third one requires m = n and yields algebras isomorphic to Mn(K). Using these families and some exceptional cases we construct all twisting maps of K 3 with K 3 .