Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A converse to a theorem of Oka and Sakamoto for complex line
  arrangements   Let C_1 and C_2 be algebraic plane curves in the complex plane such that the
curves intersect in d_1\cdot d_2 points where d_1,d_2 are the degrees of the
curves respectively. Oka and Sakamoto proved that the fundamental group of the
complement of C_1 \cup C_2 is isomorphic to the direct of product of the
fundamental group of the complement of C_1 and the fundamental group of the
complement of C_2. In this paper we prove the converse of Oka and Sakamoto's
result for line arrangements. Let A_1 and A_2 be non-empty arrangements of
lines in complex plane such that the fundamental group of the complement of A_1
\cup A_2 is isomorphic to the direct product of the complements of the
arrangements A_1 and A_2. Then, the intersection of A_1 and A_2 consists of
|A_1| \cdot |A_2| points of multiplicity two.

25
The categories of this paper are: patt-sol
14
The categories of this paper are: math-ph
9
The categories of this paper are: nlin
18
The categories of this paper are: econ
8
The categories of this paper are: hep-ex
24
The categories of this paper are: dg-ga
30
The categories of this paper are: q-alg
6
The categories of this paper are: astro-ph
7
The categories of this paper are: hep-th
29
The categories of this paper are: funct-an
1
The categories of this paper are: cs
36
The categories of this paper are: cmp-lg
26
The categories of this paper are: bayes-an
10
The categories of this paper are: q-bio
31
The categories of this paper are: ao-sci
27
The categories of this paper are: comp-gas