In the paper (2.11) the authors show that if $k^*$ is a separable algebraic extension of $k$ and $x_1,x_2, \ldots, x_n$ are indeterminates over $k^*$ and a normal one dimensional ring $A$ with $k \subset A \subset k^*[x_1,x_2,\ldots,x_n]$ then $A$ has the form $k'[t]$ where $k'$ is the algebraic closer of $k$ in $A$.
The above is a very strict sufficient condition to ensure the polynomial property. Is there a general answer to the question:
Q. When a ring $R$ is a polynomial ring over a field?