Extending the Archimedean Positivstellensatz to the Non-compact Case

A generalization of Schm udgen s Positivstellensatz is given which holds for any basic closed semialgebraic set in R compact or not The proof is an extension of W ormann s proof The Positivstellensatz proved by G Stengle in is a standard tool in real algebraic geometry see In his solution of the K moment problem in Schm udgen proves a surprisingly strong version of the Positivstellensatz in the compact case Schm udgen s result has since been extended and improved in various ways see In the present paper we describe an extension in another direction to the non compact case Let V be an algebraic set in R The coordinate ring R V of V is the ring of all polynomial functions f V R R V is generated as an R algebra by x xn where xi V R denotes the i th coordinate function For any nite subset S ff frg of R V let K KS be the basic closed semialgebraic set in V de ned by the r inequalities fi i r i e