Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Applications of Swan's Bertini to unimodular rows   Let $R$ be an affine algebra of dimension $d\geq 4$ over a perfect field $k$
of char $\neq 2$ and $I$ be an ideal of $R$. Then
  - Um$_{d+1}(R,I)/{\rm E}_{d+1}(R,I)$ has nice group structure if
$c.d._2(k)\leq 2$.
  - Um$_d(R,I)/{\rm E}_d(R,I)$ has nice group structure if $k$ is algebraically
closed of char $k\neq 2,3$ and either (i) $k = \overline{\mathbb{F}}_{p}$ or
(ii) $R$ is normal.
  - $MS_{d+1}(R)$ is uniquely divisible prime to characteristic of $k$ if $R$
is reduced and $k$ is infinite with $c.d.(k)\leq 1$.

0
The categories of this paper are: hep-ph
34
The categories of this paper are: chem-ph
19
The categories of this paper are: eess
3
The categories of this paper are: physics
1
The categories of this paper are: cs
8
The categories of this paper are: hep-ex
33
The categories of this paper are: plasm-ph
5
The categories of this paper are: gr-qc
6
The categories of this paper are: astro-ph
14
The categories of this paper are: math-ph
32
The categories of this paper are: atom-ph
10
The categories of this paper are: q-bio
12
The categories of this paper are: nucl-th
4
The categories of this paper are: cond-mat
22
The categories of this paper are: chao-dyn
7
The categories of this paper are: hep-th