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We study the LND conjecture concerning the images of locally nilpotent derivations, which arose from the Jacobian conjecture. Let be a domain containing a field of characteristic zero. We prove that, when is a one-dimensional unique factorization domain, the image of any locally nilpotent -derivation of the bivariate polynomial algebra is a Mathieu-Zhao subspace. Moreover, we prove that, when is a Dedekind domain, the image of a locally nilpotent -derivation of with some additional conditions...
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