Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: An extension of the Hermite-Biehler theorem with application to
  polynomials with one positive root   If a real polynomial $f(x)=p(x^2)+xq(x^2)$ is Hurwitz stable (every root if
$f$ lies in the open left half-plane), then the Hermite-Biehler Theorem says
that the polynomials $p(-x^2)$ and $q(-x^2)$ have interlacing real roots. We
extend this result to general polynomials by giving a lower bound on the number
of real roots of $p(-x^2)$ and $q(-x^2)$ and showing that these real roots
interlace. This bound depends on the number of roots of $f$ which lie in the
left half plane. Another classical result in the theory of polynomials is
Descartes' Rule of Signs, which bounds the number of positive roots of a
polynomial in terms of the number of sign changes in its coefficients. We use
our extension of the Hermite-Biehler Theorem to give an inverse rule of signs
for polynomials with one positive root.

12
The categories of this paper are: nucl-th
11
The categories of this paper are: hep-lat
22
The categories of this paper are: chao-dyn
30
The categories of this paper are: q-alg
37
The categories of this paper are: supr-con
21
The categories of this paper are: adap-org
7
The categories of this paper are: hep-th
25
The categories of this paper are: patt-sol
10
The categories of this paper are: q-bio
29
The categories of this paper are: funct-an
32
The categories of this paper are: atom-ph
4
The categories of this paper are: cond-mat
8
The categories of this paper are: hep-ex
6
The categories of this paper are: astro-ph
24
The categories of this paper are: dg-ga
23
The categories of this paper are: solv-int