Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A proof of hyperbolic van der Waerden conjecture : the right
  generalization is the ultimate simplification   Consider a homogeneous polynomial $p(z_1,...,z_n)$ of degree $n$ in $n$
complex variables . Assume that this polynomial satisfies the property : \\
  $|p(z_1,...,z_n)| \geq \prod_{1 \leq i \leq n} Re(z_i)$ on the domain
$\{(z_1,...,z_n) : Re(z_i) \geq 0, 1 \leq i \leq n \}$ . \\
  We prove that $|\frac{\partial^n}{\partial z_1...\partial z_n} p | \geq
\frac{n!}{n^n}$ . Our proof is relatively short and self-contained (i.e. we
only use basic properties of hyperbolic polynomials). As the van der Waerden
conjecture for permanents, proved by D.I. Falikman and G.P. Egorychev, as well
Bapat's conjecture for mixed discriminants, proved by the author, are
particular cases of this result. We also prove so called "small rank" lower
bound (in the permanents context it corresponds to sparse doubly-stochastic
matrices, i.e. with small number of non-zero entries in each column). The later
lower bound generalizes (with simpler proofs) recent lower bounds by
A.Schrijver for the number of perfect matchings of $k$-regular bipartite
graphs.
  We present some important algorithmic applications of the result, including a
polynomial time deterministic algorithm approximating the permanent of $n
\times n$ nonnegative entry-wise matrices within multiplicative factor
$\frac{e^n}{n^m}$ for any fixed positive $m$ .

9
The categories of this paper are: nlin
7
The categories of this paper are: hep-th
22
The categories of this paper are: chao-dyn
37
The categories of this paper are: supr-con
17
The categories of this paper are: q-fin
24
The categories of this paper are: dg-ga
3
The categories of this paper are: physics
11
The categories of this paper are: hep-lat
21
The categories of this paper are: adap-org
15
The categories of this paper are: nucl-ex
4
The categories of this paper are: cond-mat
16
The categories of this paper are: stat
25
The categories of this paper are: patt-sol
13
The categories of this paper are: quant-ph
34
The categories of this paper are: chem-ph
32
The categories of this paper are: atom-ph