Application of polynomials with non-negative coefficients Question 1: Are there any deeper applications (in any field of mathematics) of polynomials (with possibly more than one variable) over the real numbers whose coefficients are non-negative? So far I have found only some vague mentions of control theory and maybe some approximations of functions, but don't know the details (e.g. www.math.ttu.edu/~barnard/poly.pdf). 
Question 2: There is a similar notion, namely "positive polynomials", but they are defined as "polynomial functions that are non-negative". Does anybody know some connection between these two kinds of polynomials? Thanks.
All comments and suggested answers here (thanks for them:)) have certainly something to do with the mentioned type of polynomials. To specify more what was my intention - a desired answer (to Question 1) should fullfil the following:
"Is the application in you proposed area of such a kind that it forces an (independent) research of polynomials with non-negative coefficients on themselves?"
 A: Regarding question 1: An example from combinatorics (or conceivably topology, depending on how one classifies these things): if a finite simplicial complex is Cohen-Macaulay (defined below), then its $h$-polynomial (also defined below) has nonnegative coefficients.  So this is one example of a situation where you can detect that an object of some sort lacks some nice property by observing that an associated polynomial does not have nonnegative coefficients.  
Terminology review, as promised above: given a finite $(d-1)$-dimensional simplicial complex $\Delta $, let $f_i(\Delta )$ be the number of $i$-dimensional faces in $\Delta $, making the convention that the empty set is a $(-1)$-dimensional face.  The vector $(f_{-1}(\Delta ),f_0(\Delta ),\dots )$ is called the $f$-vector of $\Delta $.  One may encode the same data with another vector called the $h$-vector of $\Delta $ given by $h_k(\Delta ) = \sum_{i=0}^k (-1)^{k-i}{d-i\choose k-i}f_{i-1}(\Delta )$ for each $0\le k\le d$, so for instance $h_d(\Delta )$ is the reduced Euler characteristic of $\Delta $.  The $h$-polynomial of $\Delta $ is the polynomial $h(x) = h_0 x^d + h_1 x^{d-1} + \cdots + h_d $, while the $f$-polynomial of $\Delta $ is $f(x) = f_{-1}x^d + f_0x^{d-1} + \cdots + f_{d-1}$.  Now an equivalent way (to above) of describing the relationship between $h$-vectors and $f$-vectors is via $\sum_{i=0}^d f_{i-1}(t-1)^{d-i} = \sum_{i=0}^d h_it^{d-i}$, or in other words as $h(x)=f(x-1)$. 
Now a finite $d$-dimensional simplicial complex $\Delta $ is said to be Cohen-Macaulay over the integers if all its reduced homology groups satisfy $\tilde{H_i}(\Delta ,\mathbf{Z})=0$ for $i < d$ and likewise the link $lk_{\Delta }(\sigma )$ of any face  $\sigma \in \Delta $ has reduced homology groups satisfying $\tilde{H_i}(lk_{\Delta }(\sigma )) = 0$ for $i < dim (lk_{\Delta }(\sigma ))$.  Cohen-Macaulay complexes have the property that their $h$-vector is nonnegative, hence $h$-polynomial has nonnegative coefficients.  A good reference for this and related topological combinatorics is the book "Combinatorics and commutative algebra" by Richard Stanley.  
Regarding question 2: An important way of constructing positive polynomials is as sums of squares.  Many examples of positive polynomials with nonnegative coefficients may be constructed this way -- if the smaller polynomials being squared have nonnegative coefficients too.  However, not all positive polynomials are sums of squares by any means, let alone these special ones.  A good reference on this is the paper:
G. Blekherman, Nonnegative polynomials and sums of squares, Jour. Amer. Math. Soc. 25 (2012), 617-635
which proves that there are many more nonnegative polynomials than sums of squares asymptotically by showing that the volumes of cross sections of the cone of nonnegative polynomials grow much faster as total degree is increased than the corresponding cross sections of the cone of sums of squares.
A: How about Rook polynomials? http://en.wikipedia.org/wiki/Rook_polynomial
Or if you like examples in several variables, Schur polynomials, http://en.wikipedia.org/wiki/Schur_polynomial
Or a plentiful of other polynomials with combinatorial connections.
The Schur polynomials have for example applications in representation theory.
A: One answer to question 2 is Polya's theorem on forms positive on an orthant: Let a form (i.e. homogeneous polynomial) in several variables be given which is (strictly) positive whenever evaluated on non zero tuples of nonnegative reals. Then you can multiply it with a high power of the sum of the variables such that you obtain a form with all coefficients nonnegative (actually all coefficients of the "right" degree are positive).
You can also prove a lower bound on the exponent required, see:
Powers, Reznick: A new bound for Polya’s Theorem with applications to polynomials positive on polyhedra
This theorem can be used in representation theorems involving sums of squares (cf. Patricia's answer), see my article:
An algorithmic approach to Schmüdgen’s Positivstellensatz
A: Thanks to all for your comments and suggestions. The original motivation for this question was a connection between polynomials with non-negative coefficients and commutative semirings. After some search we found some papers asking for determining the least degree of a polynomial with non-negative coefficients that is divisible by a given (general) polynomial. Suprisingly similar questions were investigated repeatedly and independetly but without any deeper motivation. We deceided hence to make an overview, improvements of some results and suggested a few conjectures. The result (made before the latest updates of this webpage) can be found here http://arxiv.org/abs/1210.6868. (Suggestions and comments are welcome.)
A: If you know that the coefficients are non-negative and also integral, then the polynomial can be completely determined by the values of $p(1)$ and $p(p(1))$. There might be a way to extend this to rational coefficients, but I'm not sure.
