The complete real spectrum of a commutative ring $A$ with $1$ is introduced. Points of the complete real spectrum ${Sper}^{c}A$ are triples $\alpha =(𝔭,v,P)$, where $𝔭$ is a real prime of $A$, $v$ is a real valuation of the field $k\left(𝔭\right):=qf(A/𝔭)$ and $P$ is an ordering of the residue field of $v$. ${Sper}^{c}A$ is shown to have the structure of a spectral space in the sense of Hochster [5]. The specialization relation on ${Sper}^{c}A$ is considered. Special attention is paid to the case where the ring $A$ in question is a real holomorphy ring.

Consider a compact subset $K$ of real $n$-space defined by polynomial inequalities $g_1\ge 0,\cdots ,g_s\ge 0$. For a polynomial $f$ non-negative on $K$, natural sufficient conditions are given (in terms of first and second derivatives at the zeros of $f$ in $K$) for $f$ to have a presentation of the form $f=t_0+t_1{g}_1+\cdots +t_s{g}_s$, $t_i$ a sum of squares of polynomials. The conditions are much less restrictive than the conditions given by Scheiderer in [11, Cor. 2.6]. The proof uses Scheiderer’s main theorem in [11] as well as arguments from quadratic form theory...