Simple derivations in two variables

2024/06/03 by Anand Parkash, Parkash, Anand, Pankaj Kumar Shukla +1
Mathematics · #13C99 #13N15 #Advanced Differential Equations and Dynamical Systems #Advanced Topics in Algebra #Commutative Algebra (math.AC) #FOS: Mathematics #Nonlinear Differential Equations Analysis

paper · pdf · doi:10.48550/arxiv.2406.01019

openalex publication_date 2024/06/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let k be a field of characteristic zero. If c₁, c₂∈ k∖ \0\, s,t≥ 1 and u≥ 0, then it is shown that the k-derivations ∂_x + x^u(c₁x^ty^s+c₂)∂_y and ∂_x + x^u(c₁x^t+c₂y^s+1)∂_y of k[x,y] are simple. We also give a necessary and sufficient condition for the k-derivation y^r∂_x + (c₁x^t₁y^s₁+c₂x^t₂y^s₂)∂_y, where r, t₁, s₁, t₂, s₂ ≥ 0 and c₁, c₂∈ k, of k[x,y] to be simple.

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