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We prove an extension of the classical Real Representation Theorem (goingback to Krivine, Stone, Kadison, Dubois and Becker and often calledKadison-Dubois Theorem). It is a criterion for membership in subsemirings(sometimes called preprimes) of a commutative ring. Whereas the classicalcriterion is only applicable for functions which are positive on therepresentation space, the new criterion can under certain arithmetic conditionsbe applied also to functions which are only nonnegative. Only in the case ofpreorders (i.e., semirings containing all squares), our result follows easilyfrom recent work of Scheiderer, Kuhlmann, Marshall and Schwartz. Our proof doesnot use (and therefore shows) the classical criterion.  We illustrate the usefulness of the new criterion by deriving a theorem ofHandelman from it saying inter alia the following: If an odd power of a realpolynomial in several variables has only nonnegative coefficients, then so doall sufficiently high powers.

Certificates for nonnegativity of polynomials with zeros on compact semialgebraic sets