Mathieu-Zhao spaces over field of positive characteristic

2020/10/20 by Fengli Liu, Dan Yan, Liu, Fengli +1
Mathematics · Physics and Astronomy · #13F20 #13G99 #13N15 #14R10 #Advanced Differential Equations and Dynamical Systems #Advanced Differential Geometry Research #Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #FOS: Mathematics #Meromorphic and Entire Functions

paper · pdf · doi:10.48550/arxiv.2010.10219

openalex publication_date 2020/10/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let K be a field of characteristic p, δ a nonzero E-derivation and D=f(x₁)∂₁. We first prove that ImD is not a Mathieu-Zhao space of K[x₁] if and only if f(x₁)=x₁^rf₁(x₁^p) and r≠ 1. Then we prove that Imδ is a Mathieu-Zhao space of K[x₁] if and only if δ is not locally nilpotent. Finally, we classify some nilpotent derivations of K[x₁] and give a sufficient and necessary condition for D(I) to be a Mathieu-Zhao space of K[x₁] for any ideal I of K[x₁].

Related

• The LFED Conjecture for some E-derivations
• Mathieu-Zhao spaces of polynomial rings
• Simple derivations and their images
• Images of Linear Derivations and Linear E-derivations of K[x 1 ,x 2 ,x 3 ]
• The images of some simple derivations