We have proved the following Problem: 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 Essen : Conjecture E : Let $A:=mathbb{C}[t,u,x,y,z]$ denote a polynomial ring, and let $f(u), g(u)$ and $h(u)$ be the polynomials defined above. 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 it gurantees that the conjectures : the Cancellation Problem for affine spaces, the Linearization Problem and the Embedding Problem and the affine $mathbb{A}^n$-Fibration Problem are still open.
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