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Pfister’s theorem for varieties and reduced Witt rings. (Théorème de Pfister pour les variétés et anneaux de Witt réduits.) (French) Zbl 0601.14019
In this beautiful paper, the author gives a proof of the following theorem:
“There exist a function \(w:\mathbb N\to\mathbb N\) with the following property: for every real closed field \(R\) and every affine \(R\)-variety \(V\) with Krull dimension \(d\) the application “global signature” from the Witt ring of the coordinate ring of \(V\) to the ring of continuous functions from \(V(R)\) in \(\mathbb Z\) has a cokernel whose torsion is bounded by \(2^{w(d)}\)”.
This theorem is a quantitative generalization of a former result by the author [Math. Ann. 260, 191–210 (1982; Zbl 0507.14019)] and was known in a few particular cases (curves or complete non singular surfaces).
The idea is to rewrite the author’s cited paper and give bounds in each step of the proof.
One of the wanted bounds is given by an important result of L. Bröcker [Geom. Dedicata 16, 335–350 (1984; Zbl 0546.14016)], bounding in terms of the dimension of \(V\) the minimal number of strict inequalities required for the description of an elementary semi-algebraic open set in \(V\); the author gives a new and simplified proof of Bröcker’s theorem (section 3).
The other ingredient is a generalization, important in itself, of Pfister’s celebrated theorem “let \(R\) be a real closed field and \(L\) be a field of transcendence degree \(d\) over \(R\), then a sum of squares in \(L\) is a sum of at most \(2^ d\) squares” to the case of coordinate rings; the author’s result, proved in section 2 is the following ”let \(R\) be a real closed field and \(A\) an \(R\)-algebra without real points, with Krull dimension \(d\). Then \(-1\) is a sum of at most \(d-1+2^{d+1}\) squares”.

MSC:
14P10 Semialgebraic sets and related spaces
12D15 Fields related with sums of squares (formally real fields, Pythagorean fields, etc.)
11P05 Waring’s problem and variants
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References:
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