abstract. we have proved the following problem:let r be a c-affine domain, let t be an element in r \ c and let i : c[t ] ֒→ r be the inclusion. assume that r/t r ∼
=c[t ] c[t ]t [n−1] . then r ∼
=c c[n] .
=c c[n−1] and that rt ∼
this result leads to the negative solution of the candidate counter-example
(conjecture e) of v.arno den essen :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 a
locally nilpotent derivation on a. let ad be a subring { a ∈ a | d(a) = 0 }.
then ad ∼
6 c c[4] . consequently our result in this short paper guarantees the
=
conjectures : the cancellation problem for affine spaces”, “the linearization
problem”, “the embedding problem” and “the affine an -fibration problem”
to be still open.