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The generalized Dixmier conjecture and the two-dimensional Jacobian conjecture
We prove that the two-dimensional Jacobian conjecture $\mathrm{JC}_2$ implies the generalized Dixmier conjecture $\mathrm{DC}'_1$ for the first Weyl algebra $A_1$, which states, in one of several equivalent formulations, that every endomorphism of the two-by-two matrix ring $\operatorname{Mat}_2(A_1
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