Summary
The paper looks at certain open problems in the theory of minimal generation of semi-algebraic sets and constructible sets in the real spectrum: problems about thet-invariant and the associatedp-invariant for separating families, and about the extension of the theory to arbitrary commutative rings.
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References
Andradas, C., Bröcker, L. andRuiz, J.,Minimal generation of basic open semi-analytic sets. Invent. Math.92 (1988), 409–430.
Becker, E.,On the real spectrum of a ring and its applications to semi-algebraic geometry. Bull. Amer. Math. Soc.15 (1986), 19–60.
Bochnak, J., Coste, M. andRoy, M.-F.,Géométrie algébrique réelle. [Ergeb Math. No. (3) 12]. Springer, Berlin, Heidelberg, New York, 1987.
Bröcker, I.,Zur Theorie de quadratischen formen über formal reelen körpern. Math. Ann.210 (1974), 233–256.
Bröcker, I.,Positivbereiche in kommutativen Ringen. Abh. Math. Sem. Univ. Hamburg52 (1982), 170–178.
Bröcker, I.,Minimale Erzeugung von Positivbereichen. Geom. Dedicata16 (1984), 335–350.
Bröcker, I.,Spaces of orderings and semi-algebraic sets. InQuadratic and Hermitian forms. [Can. Math. Soc. Conf. Proc., Vol. 4]. AMS, Providence, RI, 1984, pp. 231–248.
Bröcker, I.,On separation of basic semi-algebraic sets by polynomials. Manuscripta Math.60 (1988), 497–508.
Bröcker, I.,On basic semi-algebraic sets. Exposition. Math.9 (1991), 289–334.
Bröcker, I.,On the stability index of Noetherian rings. InReal analytic and algebraic geometry, [Lecture Notes, Vol. 1420]. Springer, 1990, pp. 72–80.
Knebusch, M. andScheiderer, C.,Einführung in die reelle algebra. Vieweg, Braunschweig, Wiesbaden, 1989.
Lam, T. Y.,An introduction to real algebra. Rocky J. Math.14 (1984), 767–814.
Mahé, L.,Une démonstration élementaire du théorème de Bröcker-Scheiderer. C. R. Acad. Sci. Paris Sér. I Math.309 (1989), 613–616.
Marshall, M.,Classification of finite spaces of orderings. Can. J. Math.31 (1979), 320–330.
Marshall, M.,Spaces of orderings IV. Can. J. Math.32 (1980), 603–627.
Marshall, M.,Minimal generation of basic sets in the real spectrum of a commutative ring. Contemp. Math., to appear.
Marshall, M.,Minimal generation of constructible sets in the real spectrum of a ring. InProceedings XII Escola de Álgebra, Diamantina, Brazil, to appear.
Marshall, M.,Separating families for semi-algebraic sets. Manuscripta Math., to appear.
Marshall, M. andWalter, L.,Minimal generation of basic semi-algebraic sets over an arbitrary ordered field. InProc. Real Algebraic Geometry, Rennes. [Lecture Notes, Vol. 1524]. Springer, Berlin, 1991, pp. 346–353.
Scheiderer, C.,Stability index of real varieties. Invent. Math.97 (1989), 467–483.
Walter, L.,Quadratic forms, orderings, and quaternion algebras over rings with many units. Master's Thesis, Univ. of Sask., 1988.
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Marshall, M. On Bröcker'st-invariant and separating families for constructible sets. Aeq. Math. 48, 306–316 (1994). https://doi.org/10.1007/BF01832992
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DOI: https://doi.org/10.1007/BF01832992