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A continuous, constructive solution to Hilbert’s 17th problem. (English) Zbl 0547.12017

Let K be an ordered field, and $\overline{K}$ its unique real closure. A polynomial $F\in K\left[{x}_{1},···{x}_{n}\right]$ is called positive semi-definite (psd) if $f\ge 0$ in $\overline{K}$. Roughly speaking, Artin’s proof shows that f is a sum of squares (SOS) of rational functions ${r}_{i}\in K\left({x}_{1},···{x}_{n}\right)$ provided f is psd and each positive element of K is an SOS. More elegantly, now for all K, f is a weighted SOS: ${p}_{i}{r}_{i}^{2}$, where ${p}_{i}\in {K}^{+}$ and f is psd. During the fifties to the dependence of ${p}_{i}{r}_{i}^{2}$ on f, that is, on its variables x and coefficients c, it was given attention. The best result, by Daykin, has remained unpublished. It provides finitely many representations of f by SOS with terms ${p}_{i}{r}_{i}^{2}$ such that (i) each ${p}_{i}$ depends polynomially on c, (ii) each ${p}_{i}{r}_{k}^{2}$ is rational in both x and c, and (iii) if f is psd then, for one of the representations, all ${p}_{i}\ge 0·$

The author gives a new proof of this result, but, in contrast to Daykin, without attention to bounds on the number of terms and their degrees (depending on n and the degree of f). He also treats a natural topological variant where, in place of (ii), ${p}_{i}{r}_{i}^{2}$ depends rationally on x, and continuously (for the order topology) on both x and c. His principal result achieves this for real closed K, where ${p}_{i}$ can be absorbed (since $|x|$ is semi-algebraic and continuous), and ${r}_{i}$ is semi-algebraic in c. Having previously excluded a rational representation without Daykin’s case distinctions, he conjectures that a continuous, piecewise rational solution is possible for all K. Here the ’pieces’ are (basic) semi-algebraic sets. - As the author observed in a later preprint, the last line of p. 366 is false, with a counter-example obtained from the identity on 1.12 of p. 369. However, the false statement is not used in the paper at all, and should be omitted.

Reviewer: G.Kreisel
##### MSC:
 12J15 Ordered fields 11E10 Forms over real fields 12D15 Formally real fields 11E16 General binary quadratic forms 11E76 Forms of degree higher than two 03F55 Intuitionistic mathematics 03F65 Other constructive mathematics 11U99 Connections of number theory with logic 54H13 Topological fields, rings, etc. (topological aspects) 14Pxx Real algebraic and real analytic geometry
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