Let
k be a field and
\mathbb{V} the affine threefold in
\mathbb{A}^4_k defined by
x^m y=F(x, z, t),
m \ge 2. In this paper, we show that
\mathbb{V} \cong \mathbb{A}^3_k if and only if
f(z, t): = F(0, z, t) is a coordinate of
k[z, t]. In particular, when
k is a field of positive characteristic and
f defines a non-trivial line in the affine plane
\mathbb{A}^2_k (we shall call such a
\mathbb{V} as an Asanuma threefold), then
\mathbb{V}\ncong \mathbb{A}^3_k although
\mathbb{V} \times \mathbb{A}^1_k \cong \mathbb{A}^4_k, thereby providing a family of counter-examples to Zariski’s cancellation conjecture for the affine 3-space in positive characteristic. Our main result also proves a special case of the embedding conjecture of Abhyankar–Sathaye in arbitrary characteristic.