Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Géométrie Algébrique Réelle et Formes Quadratiques, Proc. Conf. Rennes 1981, ed. J.-L. Colliot-Thélène, M. Coste, L. Mahé, M.-F. Roy, Lect. Notes Math. 959, Springer, 1982.
Proc. Special Session on Ordered Fields and Real Algebraic Geometry, ed. D.W. Dubois, T. Recio, Contemp. Math. 8, 1982.
Real Analytic and Algebraic Geometry. Proc. Conf. Trento 1988, ed. M. Galbiati, A. Tognoli, Lect. Notes Math. 1420, Springer, 1990.
M.E. Alonso, C. Andradas: Real spectra of complete local rings. Manuscripta math. 58, 155–177 (1987).
C. Andradas: Real places in function fields. Comm. Algebra 13, 1151–1169 (1985).
: Specialization chains of real valuation rings. J. Algebra 124, 437–446 (1989).
C. Andradas, L. Bröcker, J.M. Ruiz: Minimal generation of basic open semianalytic sets. Invent. math. 92, 409–430 (1988).
J.Kr. Arason, A. Pfister: Quadratische Formen über affinen Algebren und ein algebraischer Beweis des Satzes von Borsuk-Ulam. J. reine angew. Math. 331, 181–184 (1982).
S.M. Barton: The real spectrum of higher level of a commutative ring. Ph. D. thesis, Cornell Univ., Ithaca, 1988.
E. Becker: Valuations and real places in the theory of formally real fields. In [GARFQ], pp. 1–40.
The real holomorphy ring and sums of 2n-th powers. In [GARFQ], pp. 139–181.
: Extended Artin-Schreier theory of fields. Rocky Mountain J. Math. 14, 881–897 (1984).
E. Becker, R. Berr, F. Delon, D. Gondard: Hilbert's 17-th problem for sums of 2n-th powers. Preprint 1990.
E. Becker, D. Gondard: On rings admitting orderings and 2-primary chains of orderings of higher level. Manuscripta math. 65, 63–82 (1989).
E. Becker, J. Harman, A. Rosenberg: Signatures of fields and extension theory. J. reine angew. Math. 330, 53–75 (1982).
E. Becker, B. Jacob: Rational points on algebraic varieties over a generalized real closed field: A model theoretic approach. J. reine angew. Math. 357, 77–95 (1985).
E. Becker, A. Rosenberg: Reduced forms and reduced Witt rings of higher level. J. Algebra 92, 477–503 (1985).
On the structure of reduced Witt rings of higher level. Preprint 1985 (unpublished).
E. Becker, N. Schwartz: Reduced Witt rings of infinite level. Preprint 1989.
R. Berr: Reelle algebraische Geometrie höherer Stufe. Dissertation, Ludwig-Maximilians-Universität München, 1989.
Sums of mixed powers in fields and orderings of prescribed level. Preprint 1990.
The p-real spectrum of a commutative ring. Preprint 1991.
L. Bröcker: Zur Theorie der quadratischen Formen über formal reellen Körpern. Math. Ann. 210, 233–256 (1974).
: Positivbereiche in kommutative Ringen. Abh. Math. Sem. Univ. Hamburg 52, 170–178 (1982).
: Minimale Erzeugung von Positivbereichen. Geom. Dedicata 16, 335–350 (1984).
Spaces of orderings and semialgebraic sets. In: Quadratic and Hermitian Forms, Conf. Hamilton 1983, ed. I. Hambleton, C.R. Riehm, Canad. Math. Soc. Conf. Proc. 4, 1984, pp. 231–248.
Description of semialgebraic sets by few polynomials. Lecture CIMPA, Nice 1985.
Characterization of basic semialgebraic sets. Preprint 1987.
: On the separation of basic semialgebraic sets by polynomials. Manuscripta math. 60, 497–508 (1988).
On basic semi-algebraic sets. Preprint 1988 (to appear Expos. Math.).
On the stability index of noetherian rings. In [RAAG], pp. 72–80.
L. Bröcker, H.-W. Schülting: Valuations of function fields from the geometrical point of view. J. reine angew. Math. 365, 12–32 (1986).
L. Bröcker, G. Stengle: On the Mostowski number. Math. Z. 203, 629–633 (1990).
R. Brown: Real closures of fields at orderings of higher level. Pacific J. Math. 127, 261–279 (1987).
: The behavior of chains of orderings under field extensions and places. Pacific J. Math. 127, 281–297 (1987).
G. Brumfiel: Some open problems. In [OFRAG], pp. 19–25.
M.A. Buchner, W. Kucharz: On relative real holomorphy rings. Manuscripta math. 63, 303–316 (1989).
A. Campillo, J.M. Ruiz: Some remarks on pythagorean real curve germs. J. Algebra 128, 271–275 (1990).
M.D. Choi, Z.D. Dai, T.Y. Lam, B. Reznick: The Pythagoras number of some affine algebras and local algebras. J. reine angew. Math. 336, 45–82 (1982).
M.D. Choi, T.Y. Lam, A. Prestel, B. Reznick: Sums of 2m-th powers of rational functions in one variable over real closed fields. Preprint 1991.
M.D. Choi, T.Y. Lam, B. Reznick: Even symmetric sextics. Math. Z. 195, 559–580 (1987).
M.D. Choi, T.Y. Lam, B. Reznick: Positive sextics and Schur's inequalities. J. Algebra 141, 36–77 (1991).
J.-L. Colliot-Thélène: Appendix to: K. Kato, A Hasse principle for two dimensional global fields. J. reine angew. Math. 366, 181–183 (1986).
J.-L. Colliot-Thélène, U. Jannsen: Sommes de carrés dans les corps de fonctions. C. R. Acad. Sci. Paris 312, sér. I, 759–762 (1991).
Z.D. Dai, T.Y. Lam: Levels in algebra and topology. Comment. Math. Helvetici 59, 376–424 (1984).
Z.D. Dai, T.Y. Lam, R.J. Milgram: Applications of topology to problems on sums of squares. Ens. Math. (2) 27, 277–283 (1981).
Z.D. Dai, T.Y. Lam, C.K. Peng: Levels in algebra and topology. Bull. Am. Math. Soc. (N.S.) 3, 845–848 (1980).
F. Delon: Corps et anneaux de Rolle. Proc. Am. Math. Soc. 97, 315–319 (1986).
C.N. Delzell: A finiteness theorem for open semi-algebraic sets, with applications to Hilbert's 17th problem. In [OFRAG], pp. 79–97.
Case distinctions are necessary for representing polynomials as sums of squares. In: Proc. Herbrand Symp., Logic Coll. 1981, ed. J. Stern, North Holland, 1982, pp. 87–103.
: A continuous, constructive solution to Hilbert's 17th problem. Invent. math. 76, 365–384 (1984).
M. Dickmann: The model theory of chain closed fields. J. Symb. Logic 53, 73–82 (1988).
E. Fortuna, M. Galbiati: Séparation de semi-algébriques. Geom. Dedicata 32, 211–227 (1989).
D. Gondard: Théorie du premier ordre des corps chaînables et des corps chaîne-clos. C. R. Acad. Sci. Paris 304, sér. I, 463–465 (1987).
Chainable fields and real algebraic geometry. In [RAAG], pp. 128–148.
J. Harman: Chains of higher level orderings. In [OFRAG], pp. 141–174.
B. Jacob: The model theory of generalized real closed fields. J. reine angew. Math. 323, 213–220 (1981).
P. Jaworski: Extension of orderings on fields of quotients of rings of real analytic functions. Math. Nachr. 125, 329–339 (1986).
J.L. Kleinstein, A. Rosenberg: Signatures and semisignatures of abstract Witt rings and Witt rings of semilocal rings. Canad. J. Math. 30, 872–895 (1978).
: Succinct and representational Witt rings. Pacific J. Math. 86, 99–137 (1980).
M. Knebusch: Some open problems. In: Conf. Quadratic Forms, Kingston 1976, ed. G. Orzech, Queen's Papers Pure Appl. Math. 46, 1977, pp. 361–370.
: On the local theory of signatures and reduced quadratic forms. Abh. Math. Sem. Univ. Hamburg 51, 149–195 (1981).
: An algebraic proof of the Borsuk-Ulam theorem for polynomial mappings. Proc. Am. Math. Soc. 84, 29–32 (1982).
W. Kucharz: Invertible ideals in real holomorphy rings. J. reine angew. Math. 395, 171–185 (1989).
: Sums of 2n-th powers of real meromorphic functions. Monatsh. Math. 107, 131–136 (1989).
Generating ideals in real holomorphy rings. Preprint 1989.
F.-V. Kuhlmann, A. Prestel: On places of algebraic function fields. J. reine angew. Math. 353, 181–195 (1984).
K.Y. Lam: Topological methods for studying the composition of quadratic forms. In: Quadratic and Hermitian Forms, Conf. Hamilton 1983, ed. I. Hambleton, C.R. Riehm, Canad. Math. Soc. Conf. Proc. 4, 1984, pp. 173–192.
: Some new results on composition of quadratic forms. Invent. math. 79, 467–474 (1985).
K.Y. Lam, P.Y.H. Yiu: Geometry of normed bilinear maps and the 16-square problem. Math. Ann. 284, 437–447 (1989).
T.Y. Lam, T.L. Smith: On Yuzvinsky's monomial pairings. Preprint, Berkeley 1990.
L. Mahé: Théorème de Pfister pour les variétés et anneaux de Witt réduits. Invent. math. 85, 53–72 (1986).
: Une démonstration élémentaire du théorème de Bröcker-Scheiderer. C. R. Acad. Sci. Paris 309, sér. I, 613–616 (1989).
On the geometrical stability index of a ring. Lecture, Quadratic Forms Conf., Oberwolfach 1989.
: Level and Pythagoras number of some geometric rings. Math. Z. 204, 615–629 (1990).
M. Marshall: Spaces of orderings: Systems of quadratic forms, local structure, and saturation. Comm. Algebra 12, 723–743 (1984).
Minimal generation of constructible sets in the real spectrum of a Noetherian ring. Preprint 1989.
Minimal generation of basic sets in the real spectrum of a commutative ring. Preprint 1991.
M. Marshall, C. Mulcahy: The Witt ring of a space of signatures. J. Pure Appl. Algebra 67, 179–188 (1990).
C. Mulcahy: An abstract approach to higher level forms and rigidity. Comm. Algebra 16, 577–612 (1988).
: A representation theorem for higher level reduced Witt rings. J. Algebra 119, 105–122 (1988).
J.J. Ortega: On the Pythagoras number of a real irreducible algebroid curve. Math. Ann. 289, 111–123 (1991).
A. Pfister, S. Stolz: On the level of projective spaces. Comment. Math. Helvetici 62, 286–291 (1987).
V. Powers: Characterizing reduced Witt rings of higher level. Pacific J. Math. 128, 333–347 (1987).
: Finite spaces of signatures. Canad. J. Math. 41, 808–829 (1989).
A. Prestel: Model theory of fields: An application to positive semidefinite polynomials. Mém. Soc. Math. France (N.S.) 16, 53–65 (1984).
Model theory applied to some questions about polynomials. In: Contributions to General Algebra 5, Proc. Salzburg Conf. 1986, Wien 1987, pp. 31–43.
A. Prestel, M. Bradley: Representations of a real polynomial f(X) as a sum of 2m-th powers of rational functions. In: Ordered Algebraic Structures, ed. J. Martinez, Kluwer 1989, pp. 197–207.
B. Reznick: A quantitative version of Hurwitz' theorem on the arithmetic-geometric inequality. J. reine angew. Math. 377, 108–112 (1987).
: Forms derived from the arithmetic-geometric inequality. Math. Ann. 283, 431–464 (1989).
J.M. Ruiz: A note on a separation problem. Arch. Math. 43, 422–426 (1984).
: Pythagorean real curve germs. J. Algebra 94, 126–144 (1985).
: On Hilbert's 17th problem and real Nullstellensatz for global analytic functions. Math. Z. 190, 447–454 (1985).
: Cônes locaux et complétions. C. R. Acad. Sci. Paris 302, sér. I, 67–69 (1986).
: On the real spectrum of a ring of global analytic functions. Publ. Inst. Rech. Math. Rennes 4, 84–95 (1986).
: On the connected components of a global semianalytic set. J. reine angew. Math. 392, 137–144 (1988).
: A dimension theorem for real spectra. J. Algebra 124, 271–277 (1989).
: A going-down theorem for real spectra. J. Algebra 124, 278–283 (1989).
On the topology of global semianalytic sets. In [RAAG], pp. 237–246.
: A characterization of sums of 2nth powers of global meromorphic functions. Proc. Am. Math. Soc. 109, 915–923 (1990).
The substitution test for sums of even powers. Preprint 1991.
C. Scheiderer: Stability index of real varieties. Invent. math. 97, 467–483 (1989).
J. Schmid: Eine Bemerkung zu den höheren Pythagoraszahlen reeller Körper. Manuscripta math. 61, 195–202 (1988).
H.-W. Schülting: Über die Erzeugendenanzahl invertierbarer Ideale in Prüferringen. Comm. Algebra 7, 1331–1349 (1979).
On real places of a field and their holomorphy ring. Comm. Algebra 10, 1239–1284 (1982).
Real holomography rings in real algebraic geometry. In [GARFQ], pp. 433–442.
Prime divisors on real varieties and valuation theory. J. Algebra 98, 499–514 (1986).
The binary class group of the real holomorphy ring. Prepring 1986.
N. Schwartz: Local stability and saturation in spaces of orderings. Canad. J. Math. 35, 454–477 (1983).
Chain signatures and real closures. J. reine angew. Math. 347, 1–20 (1984).
Chain signatures and higher level Witt rings. J. Algebra 110, 74–107 (1987).
D.B. Shapiro: Products of sums of squares. Expos. Math. 2, 235–261 (1984).
O. Simon: Aspects quantitatifs de Nullstellensätze et de Positivstellensätze. Nombres de Pythagore. Comm. Algebra 17, 637–667 (1989).
G. Stengle: A lower bound for the complexity of separating functions. Rocky Mountain J. Math. 14, 925–927 (1984).
A measure for semialgebraic sets related to boolean complexity. Math. Ann. 283, 203–209 (1989).
S. Stolz: The level of real projective spaces. Comment. Math. Helvetici 64, 661–674 (1989).
R.G. Swan: n-generator ideals in Prüfer domains. Pacific J. Math. 111, 433–446 (1984).
P.Y.H. Yiu: Quadratic forms between spheres and the non-existence of sums of squares formulae. Math. Proc. Camb. Phil. Soc. 100, 493–504 (1986).
Sums of squares formulae with integer coefficients. Canad. Math. Bull. 30, 318–324 (1987).
On the product of two sums of 16 squares as a sum of squares of integral bilinear forms. Quart. J. Math. Oxford (2) 41, 463–500 (1990).
S. Yuzvinsky: Orthogonal pairings of euclidean spaces. Michigan Math. J. 28, 131–145 (1981).
A series of monomial pairings. Linear Multilin. Algebra 15, 109–119 (1984).
G. Zeng: A characterization of preordered fields with the weak Hilbert property. Proc. Am. Math. Soc. 104, 335–342 (1988).
On preordered fields related to Hilbert's 17th problem. Math. Z. 206, 145–151 (1991).
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1992 Springer-Verlag
About this paper
Cite this paper
Scheiderer, C. (1992). Real algebra and its applications to geometry in the last ten years: Some major developments and results. In: Coste, M., Mahé, L., Roy, MF. (eds) Real Algebraic Geometry. Lecture Notes in Mathematics, vol 1524. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0084609
Download citation
DOI: https://doi.org/10.1007/BFb0084609
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-55992-4
Online ISBN: 978-3-540-47337-4
eBook Packages: Springer Book Archive