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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”.

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
Full Text: DOI EuDML
[1] Bochnak, J., Coste, M., Coste-Roy, M.-F.: Géométrie algébrique réelle. (à paraître)
[2] Bröcker, L.: Zur Theorie der quadratischen Formen über formal reellen Körpern. Math. Ann.210, 233-256 (1974) · Zbl 0284.13020
[3] Bröcker, L.: Real spectra and distribution of signatures. In: Geométrie algébrique réelle et formes quadratiques. Lect Notes Math.959, 249-272 (1982)
[4] Bröcker, L.: Minimale Erzeugung von Positivbereichen. Geom. Dedicata, 16 (3), 335-350 (1984) · Zbl 0546.14016
[5] Bröcker, L.: Spaces of orderings and semialgebraic sets. Can. Math. Soc., Conf. Proc.4, 231-248 (1984) · Zbl 0547.14015
[6] Choi, M.D., Dai, Z.D., Lam, T.Y., Reznick, B.: The Pythagoras number of some affine algebras and local algebras. J. Reine Angew. Math.336, 45-82 (1982) · Zbl 0523.14020
[7] Choi, M.D., Lam, T.Y., Reznick, B., Rosenberg, A.: Sums of squares in some integral domains. J. Alg.65, 234-256 (1980) · Zbl 0433.10010
[8] Choi, M.D., Knebusch, M., Lam, T.Y., Reznick, B.: Transversal zeros and positive semidefinite forms. In: Géométrie Algébrique Réelle et Formes quadratiques. Lect. Notes Math.959, 273-298 (1982)
[9] Colliot-Thélène, J.-L.: Variantes du Nullstellensatz réel et anneaux formellement réels. In: Géométrie Algébrique Réelle et Formes Quadratiques. Lect. Notes Math.959, 98-108 (1982)
[10] Colliot-Thélène, J.-L., Sansuc, J.-J.: Fibrés quadratiques et composantes connexes réelles. Math. Ann.244, 105-134 (1979) · Zbl 0418.14016
[11] Coste, M.: Ensembles semi-algébriques. In: Géométrie Algébrique Réelle et Formes Quadratiques. Lect. Notes Math.959, 109-138 (1982)
[12] Coste, M., Coste-Roy, M.-F.: Le spectre réel et la topologie des variétés algébriques sur un corps réel clos, in ?Séminaire sur la géométrie algébrique réelle?. Publ. Math. Univ. Paris VII. pp. 103-117 (1981)
[13] Coste, M., Coste-Roy, M.-F.: La topologie du spectre réel. In: Ordered Fields and Real Algebraic Geometry. Contemp. Math.8, 27-60 (1981)
[14] Dai, Z.D., Lam, T.Y.: Levels in algebra and topology. Comment. Math. Helv.59, 376-424 (1984) · Zbl 0546.10017
[15] Dietel, G.: Wittringe singulärer reeller Kurven I et II. Comm. in Alg.11 (21), 2393-2494 (1983) · Zbl 0536.14015
[16] Knebusch, M.: Symmetric bilinear forms over algebraic varieties. In: Conference on Quadratic Forms. Queen’s papers Pure Appl. Math.46, 103-283 (1977)
[17] Knebusch, M.: On algebraic curves over real closed fields I. Math. Z.150, 49-70 (1976) · Zbl 0327.14009
[18] Knebusch, M.: On the local theory of signatures and reduced quadratic forms. Abh. Math. Semin. Univ. Hamb., pp. 149-195 (1981) · Zbl 0469.10008
[19] Lam, T.Y.: The algebraic theory of quadratic forms. Reading: Benjamin 1973 · Zbl 0259.10019
[20] Lang, S.: On quasi-algebraic closure. Ann. Math.55, 373-390 (1952) · Zbl 0046.26202
[21] Mahé, L.: Signatures et composantes connexes. Math. Ann.260, 191-210 (1982) · Zbl 0507.14019
[22] Mahé, L.: Sommes de carrés et anneaux de Witt réduits. C.R. Acad. Sc. Paris 300(I)no1, 5-7 (1985) · Zbl 0579.14022
[23] Marshall, M.: The Witt ring of a space of orderings. Trans. Am. Math. Soc.258, 505-521 (1980) · Zbl 0427.10015
[24] Pfister, A.: Zur Darstellung definiter Funktionen als Summe von Quadraten. Invent. Math.4, 229-237 (1967) · Zbl 0222.10022
[25] Pfister, A.: Multiplikative quadratische Formen. Arch. Math.16, 363-370 (1965) · Zbl 0146.26001
[26] Scharlau, W.: Quadratic forms. Queen’s papers Pure Appl. Math.22, Kingston, Ontario 1969 · Zbl 0194.35104
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