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

Let K be an ordered field, and K ¯ its unique real closure. A polynomial FK[x 1 ,···x n ] is called positive semi-definite (psd) if f0 in K ¯. Roughly speaking, Artin’s proof shows that f is a sum of squares (SOS) of rational functions r i K(x 1 ,···x n ) 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 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 The best result, by Daykin, has remained unpublished.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 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:
12J15Ordered fields
11E10Forms over real fields
12D15Formally real fields
11E16General binary quadratic forms
11E76Forms of degree higher than two
03F55Intuitionistic mathematics
03F65Other constructive mathematics
11U99Connections of number theory with logic
54H13Topological fields, rings, etc. (topological aspects)
14PxxReal algebraic and real analytic geometry
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