**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?"



Application of polynomials with non-negative coefficients
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


