Note on the candidate counter-example in the cancellation problem for affine spaces posed by Arno Van den Essen

We have proved the following Problem:{it Let $R$ be a $mathbb{C}$-affine domain, let $T$ be an element in $R setminus mathbb{C}$ and let $i : mathbb{C}[T] hookrightarrow R$ be the inclusion. Assume that $R/TR cong_{mathbb{C}} mathbb{C}^{[n-1]}$ and that $R_T cong_{mathbb{C}[T]} mathbb{C}[T]_T^{[n-1]}$. Then $R cong_{mathbb{C}} mathbb{C}^{[n]}$.} This result leads to the negative solution of the candidate counter-example of V.Arno den Lessen : Conjecture E : {it Let $A:=mathbb{C}[t,u,x,y,z]$ denote a polynomial ring, and let $f(u):=u^3-3u, g(u):=u^4-4u^2$ and $h(u):=u^5-10u$ be the polynomials in $mathbb{C}[u]$. Let $D:= f(u)partial_x + g(u)partial_y + h(u)partial_z + tpartial_u$ (which is easily seen to be a locally nilpotent derivation on $A$). Then $A^D otcong_{mathbb{C}} mathbb{C}^{[4]}$.} 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 $mathbb{A}^n$-Fibration Problem are still open.