RETRACTED: Proof of the Jacobian Conjecture in the Two-Dimensional Case and Global Isochronous Centers of Polynomial Hamiltonian Differential Systems (Retracted article. See vol. 59, pg. 439, 2023)

It is proved that, up to a nonsingular linear transformation, the real Hamiltonian system <(x) over dot> = -y - P(x), <(y) over dot> = x + (y + P(x))P'(x), where P(x) is an arbitrary polynomial of degree k >= 2, is the only real polynomial Hamiltonian systems of the general form <(x) over dot> = -y - P(x, y), <(y) over dot> = x + Q(x,y), P-x'(x, y) Q(y)'(x,y), where the polynomials P(x, y) and Q(x, y) do not contain free and linear terms, that has an isochronous global center at the singular point 0(0,0). The result obtained gives a negative answer to the question posed by M. Sabatini: do there exist polynomial Hamiltonian differential systems with a Jacobian pair that have isochronous non-global centers? On the basis of these two statements, it is proved that the Jacobian conjecture is true in the two-dimensional case. Namely, all nonsingular polynomial mappings C-2 -> C-2 with constant Jacobian are exhausted by mappings represented by finitely many compositions of linear transformations and Jacobian pairs of the form (x, y + P(x)), where P(x) is an arbitrary polynomial of degree at least two.