2012/01/20 by Sususu Oda, Oda, Sususu
Mathematics · #13B25 #Advanced Differential Equations and Dynamical Systems #Advanced Topics in Algebra #Commutative Algebra (math.AC) #FOS: Mathematics #Meromorphic and Entire Functions
paper · pdf · doi:10.48550/arxiv.1201.4198
openalex publication_date 2012/01/20 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
We have proved the following Problem:\it Let R be a ℂ-affine domain, let T be an element in R ∖ ℂ and let i : ℂ[T] \hookrightarrow R be the inclusion. Assume that R/TR ≅ℂ ℂ[n-1] and that RT ≅ℂ[T] ℂ[T]T[n-1]. Then R ≅ℂ ℂ[n]. This result leads to the negative solution of the candidate counter-example of V.Arno den Lessen : Conjecture E : \it Let A:=ℂ[t,u,x,y,z] denote a polynomial ring, and let f(u):=u3-3u, g(u):=u4-4u2 and h(u):=u5-10u be the polynomials in ℂ[u]. Let D:= f'(u)∂x + g'(u)∂y + h'(u)∂z + t∂u (which is easily seen to be a locally nilpotent derivation on A). Then AD \not≅ℂ ℂ[4]. Consequently our result in this short paper guarantees that the conjectures : "the Cancellation Problem for affine spaces", "the Linearization Problem", "the Embedding Problem" and "the affine \mathbbAn-Fibration Problem" are still open.