[For the entire collection see Zbl 0601.00004.]
In 1927 Artin proved that functions which are positive definite on the real points V(R) of an affine variety V over a real closed field R are sums of squares in the function field \(F=R(V)\) of V. The proof consists of two main parts. In the first step, the sums of squares are characterized as those functions which are positive with respect to every total ordering of F. In the second part the ”center map” is studied. In the case \(R={\mathbb{R}}\) this is a map \(c: X_ F\to \bar V({\mathbb{R}})\), where \(\bar V\) denotes the projective closure of V and \(X_ F\) is the set of total orderings of F. In this paper we discuss the attempt to generalize these methods to the study of sums of 2n-th powers. Proofs apeared in J. Algebra 98, 499-514 (1986; Zbl 0595.12015).