| Description |
The Cancellation Problem for Affine Spaces
(also known as Zariski Problem) asks:
if $V$ is an affine variety over an algebraically closed field $k$
such that $V \times \A^1_k \cong \A^{n+1}_k$,
does it follow that $V \cong \A_k^n$?
Equivalently, if $A$ is an affine $k$-algebra
such that $A[X]$ is isomorphic to the polynomial ring $k[X_1, \dots,
X_{n+1}]$,
does it follow that $A$ is isomorphic to $k[X_1, \dots, X_n]$?For $n=1$, an affirmative solution to the problem was given by
S. Abhyankar, P. Eakin and W. Heinzer (1972).
For $n=2$, an affirmative solution to the problem was given by
T. Fujita (1979), M. Miyanishi and T. Sugie (1980)
in characteristic zero and by
P. Russell (1981) in positive characteristic.
Over a field $k$ of positive characteristic,
T. Asanuma (1987) constructed a three-dimensional $k$-algebra $A$
such that $A[T]$ is isomorphic to $k[X_1,X_2,X_3, X_4]$.
The example gave rise to (in the words of P. Russell)
``Asanuma's Dilemma''. For, if $A$ were isomorphic to $k[X_1, X_2,
X_3]$, then this
would give a counterexample to the Linearisation Problem over
$\A_k^3$ in positive characteristic; if not, then a counterexample to
the Cancellation Problem.
In this talk we will show that Asanuma's example $A$ is not
isomorphic to $k[X_1, X_2, X_3]$.
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