Positive polynomial

In mathematics, a positive polynomial on a particular set is a polynomial whose values are positive on that set.
Let "p" be a polynomial in "n" variables with real coefficients and let "S" be a subset of the "n"-dimensional Euclidean space ℝ"n". We say that:
For certain sets "S", there exist algebraic descriptions of all polynomials that are positive, non-negative, or zero on "S". Such a description is a positivstellensatz, nichtnegativstellensatz, or nullstellensatz. This article will focus on the former two descriptions. For the latter, see Hilbert's Nullstellensatz for the most known nullstellensatz. 
Generalizations of positivstellensatz.
Positivstellensatz also exist for trigonometric polynomials, matrix polynomials, polynomials in free variables, various quantum polynomials, etc.