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We show that the GVC (generalized vanishing conjecture) holds for the differential operator and all polynomials , where is any polynomial over the base field. The GVC arose from the study of the Jacobian conjecture.
Let be a field of characteristic zero and a -domain. Let be a retract of being the kernel of a locally nilpotent derivation of . We show that if for some principal ideal (in particular, if is a UFD), then , i.e., is a polynomial algebra over in one variable. It is natural to ask that, if a retract of a -UFD is the kernel of two commuting locally nilpotent derivations of , then does it follow that ? We give a negative answer to this question. The interest in retracts comes...
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