@article{Bedratyuk_Petravchuk_Chapovskyi_2023, title={Centralizers of linear and locally nilpotent derivations}, volume={75}, url={https://umj.imath.kiev.ua/index.php/umj/article/view/7529}, DOI={10.3842/umzh.v75i8.7529}, abstractNote={<p class="p1">UDC 512.715, 512.554.31</p> <p class="p1">Let $\mathbb{K}$ be an algebraically closed field of characteristic zero, $\mathbb{K}[x_1,\dots,x_n]$ be the polynomial algebra and $W_n(\mathbb{K})$ be the Lie algebra of all $\mathbb K$-derivations on $\mathbb{K}[x_1,\dots,x_n].$ For any derivation $D$ with linear components,<span class="Apple-converted-space"> </span>we describe the centralizer of $D$ in $W_n(\mathbb{K}),$ and<span class="Apple-converted-space"> </span>propose an algorithm for finding the generators of this centralizer regarded as a module over the ring of constants of the derivation $D$ in the case where $D$ is a basic Weitzenboeck derivation. In a more general case where a finitely generated integral domain $A$ over the field $\mathbb{K}$ is considered instead of the polynomial algebra $\mathbb{K}[x_1,\dots,x_n]$ and $D$ is a locally nilpotent derivation on $A,$ we prove that the centralizer ${\rm C}_{\rm Der} A}(D)$ of $D$ in the Lie algebra ${\rm Der} A$ of all $\mathbb K$-differentiations on $A$ is a ``large’’ subalgebra of ${\rm Der} A.$ Specifically, the rank of ${\rm C}_{\rm Der} _A}(D)$ over $A$ is equal to the transcendence degree of the field of fractions $\mathrm{Frac}(A)$ over the field~$\mathbb K.$</p>}, number={8}, journal={Ukrains’kyi Matematychnyi Zhurnal}, author={BedratyukL. and PetravchukA. and ChapovskyiIe.}, year={2023}, month={Aug.}, pages={1043 - 1052} }