Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Simple birational extensions of the polynomial ring $\C^{[3]}$   The Abhyankar-Sathaye Problem asks whether any biregular embedding of affine
spaces $A^m_k\to A^n_k$ can be rectified, that is, is equivalent to a linear
embedding up to an automorphism of the target space. Here we study this problem
for the embeddings $C^3 \to C^4$ whose image $X$ is given in $C^4$ by an
equation $p=f(x,y)u+g(x,y,z)=0$, where $f\in C[x,y],$ $f\neq 0$ and $g\in
C[x,y,z]$. Under certain additional assumptions we show that, indeed, the
polynomial $p$ is a variable of the polynomial ring $C[x,y,z,u]$ (i.e., a
coordinate of a polynomial automorphism of $C^4$). This is an analog of a
theorem due to Sathaye which concerns the case of embeddings $C^2\to C^3$.
Besides, we generalize a theorem of Miyanishi giving, for a polynomial $p$ as
above, a criterion for as when $X$ is isomorphic to $C^3$.

1
The categories of this paper are: cs
27
The categories of this paper are: comp-gas
10
The categories of this paper are: q-bio
12
The categories of this paper are: nucl-th
14
The categories of this paper are: math-ph
36
The categories of this paper are: cmp-lg
9
The categories of this paper are: nlin
21
The categories of this paper are: adap-org
33
The categories of this paper are: plasm-ph
15
The categories of this paper are: nucl-ex
13
The categories of this paper are: quant-ph
3
The categories of this paper are: physics
29
The categories of this paper are: funct-an
6
The categories of this paper are: astro-ph
35
The categories of this paper are: mtrl-th
8
The categories of this paper are: hep-ex