  We prove an extension of the classical Real Representation Theorem (going
back to Krivine, Stone, Kadison, Dubois and Becker and often called
Kadison-Dubois Theorem). It is a criterion for membership in subsemirings
(sometimes called preprimes) of a commutative ring. Whereas the classical
criterion is only applicable for functions which are positive on the
representation space, the new criterion can under certain arithmetic conditions
be applied also to functions which are only nonnegative. Only in the case of
preorders (i.e., semirings containing all squares), our result follows easily
from recent work of Scheiderer, Kuhlmann, Marshall and Schwartz. Our proof does
not use (and therefore shows) the classical criterion.
  We illustrate the usefulness of the new criterion by deriving a theorem of
Handelman from it saying inter alia the following: If an odd power of a real
polynomial in several variables has only nonnegative coefficients, then so do
all sufficiently high powers.
