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Model theory of fields: An application to positive semidefinite polynomials. (English) Zbl 0577.12018

This paper is a contribution to the problem of determining the set of 2m- th powers in a rational function field over a real closed field. Theorem: If \(f\in {\mathbb{R}}[X_ 1,...,X_ n]\) is homogeneous and positive semidefinite then f is a sum of 2m-th powers in \({\mathbb{R}}(X_ 1,...,X_ n)\) if and only if the following condition holds: (a) 2m\(| \deg f\); (b) 2m\(| ord f(p_ 1,...,p_ n)\) for \(p_ 1,...,p_ n\in {\mathbb{R}}[t]\subset {\mathbb{R}}((t))\) with ord \(p_ i=0\) for some i. (ord is the natural valuation of \({\mathbb{R}}((T)).)\)
The theorem does not extend to rational function fields over arbitrary real closed fields. For, by use of a non-standard argument, it is shown that for \(p\in {\mathbb{R}}[X]\), deg p\(=d\), p a sum of 2m-th powers in \({\mathbb{R}}(X)\) there are no bounds N and s depending solely on m and d such that there are polynomials \(g_ i\), \(h_ i\) of degree at most s and \(p=\sum^{N}_{i=1}(g_ i/h_ i)^{2m}\). An application of the compactness theorem then shows that the theorem is not true for arbitrary real closed fields.
Reviewer: N.Schwartz

MSC:

12D15 Fields related with sums of squares (formally real fields, Pythagorean fields, etc.)
12L12 Model theory of fields
12J15 Ordered fields
03C60 Model-theoretic algebra
12L15 Nonstandard arithmetic and field theory
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References:

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