2017/01/22 by Wenhua Zhao, Zhao, Wenhua · 1 citation
Biochemistry, Genetics and Molecular Biology · Mathematics · #08A35 #16D99 (secondary) #16W25 #47B47 (primary) #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry and Number Theory #Commutative Algebra (math.AC) #FOS: Mathematics #Rings and Algebras (math.RA) #Sphingolipid Metabolism and Signaling
paper · pdf · doi:10.48550/arxiv.1701.06125
openalex publication_date 2017/01/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let K be a field of characteristic zero and x a free variable. A K-\mathcal E-derivation of K[x] is a K-linear map of the form I-ϕ for some K-algebra endomorphism ϕ of K[x], where I denotes the identity map of K[x]. In this paper we study the image of an ideal of K[x] under some K-derivations and K-\mathcal E-derivations of K[x]. We show that the LFED conjecture proposed in [Z4] holds for all K-\mathcal E-derivations and all locally finite K-derivations of K[x]. We also show that the LNED conjecture proposed in [Z4] holds for all locally nilpotent K-derivations of K[x], and also for all locally nilpotent K-\mathcal E-derivations of K[x] and the ideals uK[x] such that either u=0, or deg u≤ 1, or u has at least one repeated root in the algebraic closure of K. As a bi-product, the homogeneous Mathieu subspaces (Mathieu-Zhao spaces) of the univariate polynomial algebra over an arbitrary field have also been classified.