TL;DR
This paper investigates the properties of images of simple polynomial derivations, showing they are not Mathieu-Zhao spaces, and explores the nature of locally nilpotent derivations in higher dimensions.
Contribution
It establishes that images of simple derivations are not Mathieu-Zhao spaces and proves locally nilpotent derivations are not simple in dimensions greater than one.
Findings
Images of simple Shamsuddin derivations are not Mathieu-Zhao spaces.
Images of some simple derivations in dimension three are not Mathieu-Zhao spaces.
Locally nilpotent derivations are not simple in dimension greater than one.
Abstract
In the paper, we study the relation between the images of polynomial derivations and their simplicity. We prove that the images of simple Shamsuddin derivations are not Mathieu-Zhao spaces. In addition, we also show that the images of some simple derivations in dimension three are not Mathieu-Zhao spaces. Thus, we conjecture that the images of simple derivations in dimension greater than one are not Mathieu-Zhao spaces. We also prove that locally nilpotent derivations are not simple in dimension greater than one.
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Taxonomy
TopicsAdvanced Differential Equations and Dynamical Systems · Meromorphic and Entire Functions · Advanced Topics in Algebra