Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the real roots of $\sigma$-Polynomials   The $\sigma$-polynomial is given by $\sigma(G,x) = \sum_{i=\chi(G)}^{n}
a_{i}(G)\, x^{i}$, where $a_{i}(G)$ is the number of partitions of the vertices
of $G$ into $i$ nonempty independent sets. These polynomials are closely
related to chromatic polynomials, as the chromatic polynomial of $G$ is given
by $\sum_{i=\chi(G)}^{n} a_{i}(G)\, x(x-1) \cdots (x-(i-1))$. It is known that
the closure of the real roots of chromatic polynomials is precisely $\{0,~1\}
\bigcup [32/27,\infty)$, with $(-\infty,0)$, $(0,1)$ and $(1,32/27)$ being
maximal zero-free intervals for roots of chromatic polynomials. We ask here
whether such maximal zero-free intervals exist for $\sigma$-polynomials, and
show that the only such interval is $[0,\infty)$ -- that is, the closure of the
real roots of $\sigma$-polynomials is $(-\infty,0]$.

36
The categories of this paper are: cmp-lg
18
The categories of this paper are: econ
28
The categories of this paper are: alg-geom
20
The categories of this paper are: acc-phys
26
The categories of this paper are: bayes-an
23
The categories of this paper are: solv-int
31
The categories of this paper are: ao-sci
22
The categories of this paper are: chao-dyn
25
The categories of this paper are: patt-sol
13
The categories of this paper are: quant-ph
33
The categories of this paper are: plasm-ph
10
The categories of this paper are: q-bio
16
The categories of this paper are: stat
17
The categories of this paper are: q-fin
15
The categories of this paper are: nucl-ex
14
The categories of this paper are: math-ph