Images of locally finite \(\mathcal{E} \)-derivations of bivariate polynomial algebras (Q6105009)

The study of derivations images is motivated by their connection to the Jacobian conjecture. A Mathieu subspace, also known as a Mathieu-Zhao subspace, refers to a subspace of a commutative \(K\)-algebra \(A\) meeting a specific condition. Generally, verifying the existence of a Mathieu subspace is challenging, and even in the case of univariate polynomial algebras, Mathieu subspaces have yet to be completely determined. However, \textit{A. van den Essen} et al. [J. Pure Appl. Algebra 215, No. 9, 2130--2134 (2011; Zbl 1229.13022)] established an equivalence between the Jacobian conjecture for two dimensions and the statement that the image \(\mathrm{Im} D\) is a Mathieu subspace of \(K[x, y]\) for any \(K\)-derivation \(D\) of \(K[x,y]\) that satisfies certain conditions. They prove that if \(D\) is a locally finite \(K\)-derivation, then \(\mathrm{Im}(D)\) is a Mathieu subspace. Furthermore, Zhao also considered \(\mathcal{E}\)-derivations. A \(K\)-\(\mathcal{E}\)-derivation of a \(K\)-algebra is a linear map \(\delta\) such that \(\mathrm{id}\)-\(\delta\) is an algebra homomorphism. Zhao formulated the LFED conjecture, which states that images of locally finite \(K\)-derivations and \(K\)-\(\mathcal{E}\)-derivations are Mathieu subspaces. Previous work by van den Essen, Wright, and Zhao proved this conjecture for derivations. This paper focuses specifically on \(\mathcal{E}\)-derivations for polynomial algebras with two variables. The authors classify locally finite endomorphisms of \(\mathbb C[x, y]\) and prove the LFED conjecture for \(\mathcal{E}\)-derivations in this context. Thus, this result, together with that of van den Essen, Wright, and Zhao, confirms the LFED conjecture in the case of polynomial algebras with two variables.