On the Archimedean Positivstellensatz in Real Algebraic Geometry

A variant of the Archimedean Positivstellensatz is proved which is based on Archimedean semirings or quadratic modules of generating subalgebras. It allows one to obtain representations of strictly positive polynomials on compact semi-algebraic by means of smaller sets of squares or polynomials. A large number of examples is developed in detail.