# Commutative and Noncommutative Positivstellensaetze

Artin's solution of Hilbert's 17th problem implies that for each polynomial $p\in R[x_1,\dots,x_d]$ which is nonnegative on $R^d$ there is a polynomial $q\neq 0$ such that $q^2p$ is a sum of squares of polynomials. Positivstellensaetze can be considered as generalizations of this classical result. They characterize polynomials that are positive or nonnegative on subsets of $R^d$ which are described by finitely many polynomial inequalities. In the first half of the talk we review basic Positivstellensaetze of (commutative) real algebraic geometry by including a few recent results. The second half of the talk is concerned with Positivstellensaetze for noncommutative $\ast$-algebras. We present a Positivstellensatz for the Weyl algebra and briefly discuss various generalizations of Artin's theorem to noncommutative $\ast$--algebras.