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Real algebraic geometry

March 06, 2023

In mathematics, real algebraic geometry is the sub-branch of algebraic geometry studying real algebraic sets, i.e. real-number solutions to algebraic equations with real-number coefficients, and mappings between them (in particular real polynomial mappings).

Semialgebraic geometry is the study of semialgebraic sets, i.e. real-number solutions to algebraic inequalities with-real number coefficients, and mappings between them. The most natural mappings between semialgebraic sets are semialgebraic mappings, i.e., mappings whose graphs are semialgebraic sets.

Terminology

Nowadays the words 'semialgebraic geometry' and 'real algebraic geometry' are used as synonyms, because real algebraic sets cannot be studied seriously without the use of semialgebraic sets. For example, a projection of a real algebraic set along a coordinate axis need not be a real algebraic set, but it is always a semialgebraic set: this is the Tarski–Seidenberg theorem.[1][2] Related fields are o-minimal theory and real analytic geometry.

Examples: Real plane curves are examples of real algebraic sets and polyhedra are examples of semialgebraic sets. Real algebraic functions and Nash functions are examples of semialgebraic mappings. Piecewise polynomial mappings (see the Pierce–Birkhoff conjecture) are also semialgebraic mappings.

Computational real algebraic geometry is concerned with the algorithmic aspects of real algebraic (and semialgebraic) geometry. The main algorithm is cylindrical algebraic decomposition. It is used to cut semialgebraic sets into nice pieces and to compute their projections.

Real algebra is the part of algebra which is relevant to real algebraic (and semialgebraic) geometry. It is mostly concerned with the study of ordered fields and ordered rings (in particular real closed fields) and their applications to the study of positive polynomials and sums-of-squares of polynomials. (See Hilbert's 17th problem and Krivine's Positivestellensatz.) The relation of real algebra to real algebraic geometry is similar to the relation of commutative algebra to complex algebraic geometry. Related fields are the theory of moment problems, convex optimization, the theory of quadratic forms, valuation theory and model theory.

Timeline of real algebra and real algebraic geometry

  • 1826 Fourier's algorithm for systems of linear inequalities.[3] Rediscovered by Lloyd Dines in 1919[4] and Theodore Motzkin in 1936.[5]
  • 1835 Sturm's theorem on real root counting[6]
  • 1856 Hermite's theorem on real root counting.[7]
  • 1876 Harnack's curve theorem.[8] (This bound on the number of components was later extended to all Betti numbers of all real algebraic sets[9][10][11] and all semialgebraic sets.[12])
  • 1888 Hilbert's theorem on ternary quartics.[13]
  • 1900 Hilbert's problems (especially the 16th and the 17th problem)
  • 1902 Farkas' lemma[14] (Can be reformulated as linear positivstellensatz.)
  • 1914 Annibale Comessatti showed that not every real algebraic surface is birational to RP2[15]
  • 1916 Fejér's conjecture about nonnegative trigonometric polynomials.[16] (Solved by Frigyes Riesz.[17])
  • 1927 Emil Artin's solution of Hilbert's 17th problem[18]
  • 1927 Krull–Baer Theorem[19][20] (connection between orderings and valuations)
  • 1928 Pólya's Theorem on positive polynomials on a simplex[21]
  • 1929 B. L. van der Waerden sketches a proof that real algebraic and semialgebraic sets are triangularizable,[22] but the necessary tools have not been developed to make the argument rigorous.
  • 1931 Alfred Tarski's real quantifier elimination.[23] Improved and popularized by Abraham Seidenberg in 1954.[24] (Both use Sturm's theorem.)
  • 1936 Herbert Seifert proved that every closed smooth submanifold of   with trivial normal bundle, can be isotoped to a component of a nonsingular real algebraic subset of   which is a complete intersection[25] (from the conclusion of this theorem the word "component" can not be removed[26]).
  • 1940 Marshall Stone's representation theorem for partially ordered rings.[27] Improved by Richard Kadison in 1951[28] and Donald Dubois in 1967[29] (Kadison–Dubois representation theorem). Further improved by Mihai Putinar in 1993[30] and Jacobi in 2001[31] (Putinar–Jacobi representation theorem).
  • 1952 John Nash proved that every closed smooth manifold is diffeomorphic to a nonsingular component of a real algebraic set.[32]
  • 1956 Pierce–Birkhoff conjecture formulated.[33] (Solved in dimensions ≤ 2.[34])
  • 1964 Krivine's Nullstellensatz and Positivestellensatz.[35] Rediscovered and popularized by Stengle in 1974.[36] (Krivine uses real quantifier elimination while Stengle uses Lang's homomorphism theorem.[37])
  • 1964 Lojasiewicz triangulated semi-analytic sets[38]
  • 1964 Heisuke Hironaka proved the resolution of singularity theorem[39]
  • 1964 Hassler Whitney proved that every analytic variety admits a stratification satisfying the Whitney conditions.[40]
  • 1967 Theodore Motzkin finds a positive polynomial which is not a sum of squares of polynomials.[41]
  • 1972 Vladimir Rokhlin proved Gudkov's conjecture.[42]
  • 1973 Alberto Tognoli proved that every closed smooth manifold is diffeomorphic to a nonsingular real algebraic set.[43]
  • 1975 George E. Collins discovers cylindrical algebraic decomposition algorithm, which improves Tarski's real quantifier elimination and allows to implement it on a computer.[44]
  • 1973 Jean-Louis Verdier proved that every subanalytic set admits a stratification with condition (w).[45]
  • 1979 Michel Coste and Marie-Françoise Roy discover the real spectrum of a commutative ring.[46]
  • 1980 Oleg Viro introduced the "patch working" technique and used it to classify real algebraic curves of low degree.[47] Later Ilya Itenberg and Viro used it to produce counterexamples to the Ragsdale conjecture,[48][49] and Grigory Mikhalkin applied it to tropical geometry for curve counting.[50]
  • 1980 Selman Akbulut and Henry C. King gave a topological characterization of real algebraic sets with isolated singularities, and topologically characterized nonsingular real algebraic sets (not necessarily compact)[51]
  • 1980 Akbulut and King proved that every knot in   is the link of a real algebraic set with isolated singularity in  [52]
  • 1981 Akbulut and King proved that every compact PL manifold is PL homeomorphic to a real algebraic set.[53][54][55]
  • 1983 Akbulut and King introduced "Topological Resolution Towers" as topological models of real algebraic sets, from this they obtained new topological invariants of real algebraic sets, and topologically characterized all 3-dimensional algebraic sets.[56] These invariants later generalized by Michel Coste and Krzysztof Kurdyka[57] as well as Clint McCrory and Adam Parusiński.[58]
  • 1984 Ludwig Bröcker's theorem on minimal generation of basic open semialgebraic sets[59] (improved and extended to basic closed semialgebraic sets by Scheiderer.[60])
  • 1984 Benedetti and Dedo proved that not every closed smooth manifold is diffeomorphic to a totally algebraic nonsingular real algebraic set (totally algebraic means all its Z/2Z-homology cycles are represented by real algebraic subsets).[61]
  • 1991 Akbulut and King proved that every closed smooth manifold is homeomorphic to a totally algebraic real algebraic set.[62]
  • 1991 Schmüdgen's solution of the multidimensional moment problem for compact semialgebraic sets and related strict positivstellensatz.[63] Algebraic proof found by Wörmann.[64] Implies Reznick's version of Artin's theorem with uniform denominators.[65]
  • 1992 Akbulut and King proved ambient versions of the Nash-Tognoli theorem: Every closed smooth submanifold of Rn is isotopic to the nonsingular points (component) of a real algebraic subset of Rn, and they extended this result to immersed submanifolds of Rn.[66][67]
  • 1992 Benedetti and Marin proved that every compact closed smooth 3-manifold M can be obtained from   by a sequence of blow ups and downs along smooth centers, and that M is homeomorphic to a possibly singular affine real algebraic rational threefold[68]
  • 1997 Bierstone and Milman proved a canonical resolution of singularities theorem[69]
  • 1997 Mikhalkin proved that every closed smooth n-manifold can be obtained from   by a sequence of topological blow ups and downs[70]
  • 1998 János Kollár showed that not every closed 3-manifold is a projective real 3-fold which is birational to RP3[71]
  • 2000 Scheiderer's local-global principle and related non-strict extension of Schmüdgen's positivstellensatz in dimensions ≤ 2.[72][73][74]
  • 2000 János Kollár proved that every closed smooth 3–manifold is the real part of a compact complex manifold which can be obtained from   by a sequence of real blow ups and blow downs.[75]
  • 2003 Welschinger introduces an invariant for counting real rational curves[76]
  • 2005 Akbulut and King showed that not every nonsingular real algebraic subset of RPn is smoothly isotopic to the real part of a nonsingular complex algebraic subset of CPn[77][78]

References

  • S. Akbulut and H.C. King, Topology of real algebraic sets, MSRI Pub, 25. Springer-Verlag, New York (1992) ISBN 0-387-97744-9
  • Bochnak, Jacek; Coste, Michel; Roy, Marie-Françoise. Real Algebraic Geometry. Translated from the 1987 French original. Revised by the authors. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 36. Springer-Verlag, Berlin, 1998. x+430 pp. ISBN 3-540-64663-9
  • Basu, Saugata; Pollack, Richard; Roy, Marie-Françoise Algorithms in real algebraic geometry. Second edition. Algorithms and Computation in Mathematics, 10. Springer-Verlag, Berlin, 2006. x+662 pp. ISBN 978-3-540-33098-1; 3-540-33098-4
  • Marshall, Murray Positive polynomials and sums of squares. Mathematical Surveys and Monographs, 146. American Mathematical Society, Providence, RI, 2008. xii+187 pp. ISBN 978-0-8218-4402-1; 0-8218-4402-4

Notes

  1. ^ van den Dries, L. (1998). Tame topology and o-minimal structures. London Mathematical Society Lecture Note Series. Vol. 248. Cambridge University Press. p. 31. Zbl 0953.03045.
  2. ^ Khovanskii, A. G. (1991). Fewnomials. Translations of Mathematical Monographs. Vol. 88. Translated from the Russian by Smilka Zdravkovska. Providence, RI: American Mathematical Society. ISBN 0-8218-4547-0. Zbl 0728.12002.
  3. ^ Joseph B. J. Fourier, Solution d'une question particuliére du calcul des inégalités. Bull. sci. Soc. Philomn. Paris 99–100. OEuvres 2, 315–319.
  4. ^ Dines, Lloyd L. (1919). "Systems of linear inequalities". Annals of Mathematics. (2). 20 (3): 191–199. doi:10.2307/1967869. JSTOR 1967869.
  5. ^ Theodore Motzkin, Beiträge zur Theorie der linearen Ungleichungen. IV+ 76 S. Diss., Basel (1936).
  6. ^ Jacques Charles François Sturm, Mémoires divers présentés par des savants étrangers 6, pp. 273–318 (1835).
  7. ^ Charles Hermite, Sur le Nombre des Racines d’une Équation Algébrique Comprise Entre des Limites Données, Journal für die reine und angewandte Mathematik, vol. 52, pp. 39–51 (1856).
  8. ^ C. G. A. Harnack Über Vieltheiligkeit der ebenen algebraischen Curven, Mathematische Annalen 10 (1876), 189–199
  9. ^ I. G. Petrovski˘ı and O. A. Ole˘ınik, On the topology of real algebraic surfaces, Izvestiya Akad. Nauk SSSR. Ser.Mat. 13, (1949). 389–402
  10. ^ John Milnor, On the Betti numbers of real varieties, Proceedings of the American Mathematical Society 15 (1964), 275–280.
  11. ^ René Thom, Sur l’homologie des vari´et´es algebriques r´eelles, in: S. S. Cairns (ed.), Differential and Combinatorial Topology, pp. 255–265, Princeton University Press, Princeton, NJ, 1965.
  12. ^ Basu, Saugata (1999). "On bounding the Betti numbers and computing the Euler characteristic of semi-algebraic sets". Discrete & Computational Geometry. 22 (1): 1–18. doi:10.1007/PL00009443. S2CID 7023328.
  13. ^ Hilbert, David (1888). "Uber die Darstellung definiter Formen als Summe von Formenquadraten". Mathematische Annalen. 32 (3): 342–350. doi:10.1007/BF01443605. S2CID 177804714.
  14. ^ Farkas, Julius. "Über die Theorie der Einfachen Ungleichungen". Journal für die Reine und Angewandte Mathematik. 124: 1–27.
  15. ^ Comessatti, Annibale (1914). "Sulla connessione delle superfizie razionali reali". Annali di Math. 23 (3): 215–283. doi:10.1007/BF02419577. S2CID 121297483.
  16. ^ Lipót Fejér, ¨Uber trigonometrische Polynome, J. Reine Angew. Math. 146 (1916), 53–82.
  17. ^ Frigyes Riesz and Béla Szőkefalvi-Nagy, Functional Analysis, Frederick Ungar Publ. Co., New York, 1955.
  18. ^ Artin, Emil (1927). "Uber die Zerlegung definiter Funktionen in Quadrate". Abh. Math. Sem. Univ. Hamburg. 5: 85–99. doi:10.1007/BF02952512. S2CID 122881707.
  19. ^ Krull, Wolfgang (1932). "Allgemeine Bewertungstheorie". Journal für die reine und angewandte Mathematik. 1932 (167): 160–196. doi:10.1515/crll.1932.167.160. S2CID 199547002.
  20. ^ Baer, Reinhold (1927), "Über nicht-archimedisch geordnete Körper", Sitzungsberichte der Heidelberger Akademie der Wissenschaften. Mathematisch-Naturwissenschaftliche Klasse, 8: 3–13
  21. ^ George Pólya, Über positive Darstellung von Polynomen Vierteljschr, Naturforsch. Ges. Zürich 73 (1928) 141–145, in: R.P. Boas (Ed.), Collected Papers Vol. 2, MIT Press, Cambridge, MA, 1974, pp. 309–313
  22. ^ B. L. van der Waerden, Topologische Begründung des Kalküls der abzählenden Geometrie. Math. Ann. 102, 337–362 (1929).
  23. ^ Alfred Tarski, A decision method for elementary algebra and geometry, Rand. Corp.. 1948; UC Press, Berkeley, 1951, Announced in : Ann. Soc. Pol. Math. 9 (1930, published 1931) 206–7; and in Fund. Math. 17 (1931) 210–239.
  24. ^ Abraham Seidenberg, A new decision method for elementary algebra, Annals of Mathematics 60 (1954), 365–374.
  25. ^ Herbert Seifert, Algebraische approximation von Mannigfaltigkeiten, Mathematische Zeitschrift 41 (1936), 1–17
  26. ^ Selman Akbulut and Henry C. King, Submanifolds and homology of nonsingular real algebraic varieties, American Journal of Mathematics, vol. 107, no. 1 (Feb., 1985) p.72
  27. ^ Stone, Marshall (1940). "A general theory of spectra. I." Proceedings of the National Academy of Sciences of the United States of America. 26 (4): 280–283. doi:10.1073/pnas.26.4.280. PMC 1078172. PMID 16588355.
  28. ^ Kadison, Richard V. (1951), "A representation theory for commutative topological algebra", Memoirs of the American Mathematical Society, 7: 39 pp, MR 0044040
  29. ^ Dubois, Donald W. (1967). "A note on David Harrison's theory of preprimes". Pacific Journal of Mathematics. 21: 15–19. doi:10.2140/pjm.1967.21.15. MR 0209200.
  30. ^ Mihai Putinar, Positive polynomials on compact semi-algebraic sets. Indiana University Mathematics Journal 42 (1993), no. 3, 969–984.
  31. ^ T. Jacobi, A representation theorem for certain partially ordered commutative rings. Mathematische Zeitschrift 237 (2001), no. 2, 259–273.
  32. ^ Nash, John (1952). "Real algebraic manifolds". Annals of Mathematics. 56: 405–421. doi:10.2307/1969649. JSTOR 1969649.
  33. ^ Birkhoff, Garrett; Pierce, Richard Scott (1956). "Lattice ordered rings". Anais da Academia Brasileira de Ciências. 28: 41–69.
  34. ^ Mahé, Louis (1984). "On the Pierce–Birkhoff conjecture". Rocky Mountain Journal of Mathematics. 14 (4): 983–985. doi:10.1216/RMJ-1984-14-4-983. MR 0773148.
  35. ^ Krivine, J.-L. (1964). "Anneaux préordonnés" (PDF). Journal d'Analyse Mathématique. 12: 307–326. doi:10.1007/BF02807438.
  36. ^ G. Stengle, A nullstellensatz and a positivstellensatz in semialgebraic geometry. Math. Ann. 207 (1974), 87–97.
  37. ^ S. Lang, Algebra. Addison–Wesley Publishing Co., Inc., Reading, Mass. 1965 xvii+508 pp.
  38. ^ S. Lojasiewicz, Triangulation of semi-analytic sets, Ann. Scu. Norm. di Pisa, 18 (1964), 449–474.
  39. ^ Heisuke Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero. I, Annals of Mathematics (2) 79 (1): (1964) 109–203, and part II, pp. 205–326.
  40. ^ Hassler Whitney, Local properties of analytic varieties, Differential and combinatorial topology (ed. S. Cairns), Princeton Univ. Press, Princeton N.J. (1965), 205–244.
  41. ^ Theodore S. Motzkin, The arithmetic-geometric inequality. 1967 Inequalities (Proc. Sympos. Wright-Patterson Air Force Base, Ohio, 1965) pp. 205–224 MR0223521.
  42. ^ "Proof of Gudkov's hypothesis". V. A. Rokhlin. Functional Analysis and Its Applications, volume 6, pp. 136–138 (1972)
  43. ^ Alberto Tognoli, Su una congettura di Nash, Annali della Scuola Normale Superiore di Pisa 27, 167–185 (1973).
  44. ^ George E. Collins, "Quantifier elimination for real closed fields by cylindrical algebraic decomposition", Lect. Notes Comput. Sci. 33, 134–183, 1975 MR0403962.
  45. ^ Jean-Louis Verdier, Stratifications de Whitney et théorème de Bertini-Sard, Inventiones Mathematicae 36, 295–312 (1976).
  46. ^ Marie-Françoise Coste-Roy, Michel Coste, Topologies for real algebraic geometry. Topos theoretic methods in geometry, pp. 37–100, Various Publ. Ser., 30, Aarhus Univ., Aarhus, 1979.
  47. ^ Oleg Ya. Viro, Gluing of plane real algebraic curves and constructions of curves of degrees 6 and 7. In Topology (Leningrad, 1982), volume 1060 of Lecture Notes in Mathematics, pages 187–200. Springer, Berlin, 1984
  48. ^ Viro, Oleg Ya. (1980). "Кривые степени 7, кривые степени 8 и гипотеза Рэгсдейл" [Curves of degree 7, curves of degree 8 and the hypothesis of Ragsdale]. Doklady Akademii Nauk SSSR. 254 (6): 1306–1309. Translated in "Curves of degree 7, curves of degree 8 and Ragsdale's conjecture". Soviet Mathematics - Doklady. 22: 566–570. 1980. Zbl 0422.14032.
  49. ^ Itenberg, Ilia; Mikhalkin, Grigory; Shustin, Eugenii (2007). Tropical algebraic geometry. Oberwolfach Seminars. Vol. 35. Basel: Birkhäuser. pp. 34–35. ISBN 978-3-7643-8309-1. Zbl 1162.14300.
  50. ^ Mikhalkin, Grigory (2005). "Enumerative tropical algebraic geometry in  ". Journal of the American Mathematical Society. 18: 313–377. doi:10.1090/S0894-0347-05-00477-7.
  51. ^ Selman Akbulut and Henry C. King, The topology of real algebraic sets with isolated singularities, Annals of Mathematics 113 (1981), 425–446.
  52. ^ Selman Akbulut and Henry C. King, All knots are algebraic, Commentarii Mathematici Helvetici 56, Fasc. 3 (1981), 339–351.
  53. ^ S. Akbulut and H.C. King, Real algebraic structures on topological spaces, Publications Mathématiques de l'IHÉS 53 (1981), 79–162.
  54. ^ S. Akbulut and L. Taylor, A topological resolution theorem, Publications Mathématiques de l'IHÉS 53 (1981), 163–196.
  55. ^ S. Akbulut and H.C. King, The topology of real algebraic sets, L'Enseignement Mathématique 29 (1983), 221–261.
  56. ^ Selman Akbulut and Henry C. King, Topology of real algebraic sets, MSRI Pub, 25. Springer-Verlag, New York (1992) ISBN 0-387-97744-9
  57. ^ Coste, Michel; Kurdyka, Krzysztof (1992). "On the link of a stratum in a real algebraic set". Topology. 31 (2): 323–336. doi:10.1016/0040-9383(92)90025-d. MR 1167174.
  58. ^ McCrory, Clint; Parusiński, Adam (2007), "Algebraically constructible functions: real algebra and topology", Arc spaces and additive invariants in real algebraic and analytic geometry, Panoramas et Synthèses, vol. 24, Paris: Société mathématique de France, pp. 69–85, arXiv:math/0202086, MR 2409689
  59. ^ Bröcker, Ludwig (1984). "Minimale erzeugung von Positivbereichen". Geometriae Dedicata (in German). 16 (3): 335–350. doi:10.1007/bf00147875. MR 0765338. S2CID 117475206.
  60. ^ C. Scheiderer, Stability index of real varieties. Inventiones Mathematicae 97 (1989), no. 3, 467–483.
  61. ^ R. Benedetti and M. Dedo, Counterexamples to representing homology classes by real algebraic subvarieties up to homeomorphism, Compositio Mathematica, 53, (1984), 143–151.
  62. ^ S. Akbulut and H.C. King, All compact manifolds are homeomorphic to totally algebraic real algebraic sets, Comment. Math. Helvetici 66 (1991) 139–149.
  63. ^ K. Schmüdgen, The K-moment problem for compact semi-algebraic sets. Math. Ann. 289 (1991), no. 2, 203–206.
  64. ^ T. Wörmann Strikt Positive Polynome in der Semialgebraischen Geometrie, Univ. Dortmund 1998.
  65. ^ B. Reznick, Uniform denominators in Hilbert's seventeenth problem. Math. Z. 220 (1995), no. 1, 75–97.
  66. ^ S. Akbulut and H.C. King On approximating submanifolds by algebraic sets and a solution to the Nash conjecture, Inventiones Mathematicae 107 (1992), 87–98
  67. ^ S. Akbulut and H.C. King, Algebraicity of Immersions, Topology, vol. 31, no. 4, (1992), 701–712.
  68. ^ R. Benedetti and A. Marin , Déchirures de variétés de dimension trois ...., Comm. Math. Helv. 67 (1992), 514–545.
  69. ^ E. Bierstone and P.D. Milman , Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant, Inventiones Mathematicae 128 (2) (1997) 207–302.
  70. ^ G. Mikhalkin, Blow up equivalence of smooth closed manifolds, Topology, 36 (1997) 287–299
  71. ^ János Kollár, The Nash conjecture for algebraic threefolds, ERA of AMS 4 (1998) 63–73
  72. ^ C. Scheiderer, Sums of squares of regular functions on real algebraic varieties. Transactions of the American Mathematical Society 352 (2000), no. 3, 1039–1069.
  73. ^ C. Scheiderer, Sums of squares on real algebraic curves, Mathematische Zeitschrift 245 (2003), no. 4, 725–760.
  74. ^ C. Scheiderer, Sums of squares on real algebraic surfaces. Manuscripta Mathematica 119 (2006), no. 4, 395–410.
  75. ^ János Kollár, The Nash conjecture for nonprojective threefolds, arXiv:math/0009108v1
  76. ^ J.-Y. Welschinger, Invariants of real rational symplectic 4-manifolds and lower bounds in real enumerative geometry, Inventiones Mathematicae 162 (2005), no. 1, 195–234. Zbl 1082.14052
  77. ^ S. Akbulut and H.C. King, Transcendental submanifolds of RPn Comm. Math. Helv., 80, (2005), 427–432
  78. ^ S. Akbulut, Real algebraic structures, Proceedings of GGT, (2005) 49–58, arXiv:math/0601105v3.

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March 06, 2023
real, algebraic, geometry, mathematics, real, algebraic, geometry, branch, algebraic, geometry, studying, real, algebraic, sets, real, number, solutions, algebraic, equations, with, real, number, coefficients, mappings, between, them, particular, real, polynom. In mathematics real algebraic geometry is the sub branch of algebraic geometry studying real algebraic sets i e real number solutions to algebraic equations with real number coefficients and mappings between them in particular real polynomial mappings Semialgebraic geometry is the study of semialgebraic sets i e real number solutions to algebraic inequalities with real number coefficients and mappings between them The most natural mappings between semialgebraic sets are semialgebraic mappings i e mappings whose graphs are semialgebraic sets Contents 1 Terminology 2 Timeline of real algebra and real algebraic geometry 3 References 4 Notes 5 External linksTerminology EditNowadays the words semialgebraic geometry and real algebraic geometry are used as synonyms because real algebraic sets cannot be studied seriously without the use of semialgebraic sets For example a projection of a real algebraic set along a coordinate axis need not be a real algebraic set but it is always a semialgebraic set this is the Tarski Seidenberg theorem 1 2 Related fields are o minimal theory and real analytic geometry Examples Real plane curves are examples of real algebraic sets and polyhedra are examples of semialgebraic sets Real algebraic functions and Nash functions are examples of semialgebraic mappings Piecewise polynomial mappings see the Pierce Birkhoff conjecture are also semialgebraic mappings Computational real algebraic geometry is concerned with the algorithmic aspects of real algebraic and semialgebraic geometry The main algorithm is cylindrical algebraic decomposition It is used to cut semialgebraic sets into nice pieces and to compute their projections Real algebra is the part of algebra which is relevant to real algebraic and semialgebraic geometry It is mostly concerned with the study of ordered fields and ordered rings in particular real closed fields and their applications to the study of positive polynomials and sums of squares of polynomials See Hilbert s 17th problem and Krivine s Positivestellensatz The relation of real algebra to real algebraic geometry is similar to the relation of commutative algebra to complex algebraic geometry Related fields are the theory of moment problems convex optimization the theory of quadratic forms valuation theory and model theory Timeline of real algebra and real algebraic geometry Edit1826 Fourier s algorithm for systems of linear inequalities 3 Rediscovered by Lloyd Dines in 1919 4 and Theodore Motzkin in 1936 5 1835 Sturm s theorem on real root counting 6 1856 Hermite s theorem on real root counting 7 1876 Harnack s curve theorem 8 This bound on the number of components was later extended to all Betti numbers of all real algebraic sets 9 10 11 and all semialgebraic sets 12 1888 Hilbert s theorem on ternary quartics 13 1900 Hilbert s problems especially the 16th and the 17th problem 1902 Farkas lemma 14 Can be reformulated as linear positivstellensatz 1914 Annibale Comessatti showed that not every real algebraic surface is birational to RP2 15 1916 Fejer s conjecture about nonnegative trigonometric polynomials 16 Solved by Frigyes Riesz 17 1927 Emil Artin s solution of Hilbert s 17th problem 18 1927 Krull Baer Theorem 19 20 connection between orderings and valuations 1928 Polya s Theorem on positive polynomials on a simplex 21 1929 B L van der Waerden sketches a proof that real algebraic and semialgebraic sets are triangularizable 22 but the necessary tools have not been developed to make the argument rigorous 1931 Alfred Tarski s real quantifier elimination 23 Improved and popularized by Abraham Seidenberg in 1954 24 Both use Sturm s theorem 1936 Herbert Seifert proved that every closed smooth submanifold of R n displaystyle mathbb R n with trivial normal bundle can be isotoped to a component of a nonsingular real algebraic subset of R n displaystyle mathbb R n which is a complete intersection 25 from the conclusion of this theorem the word component can not be removed 26 1940 Marshall Stone s representation theorem for partially ordered rings 27 Improved by Richard Kadison in 1951 28 and Donald Dubois in 1967 29 Kadison Dubois representation theorem Further improved by Mihai Putinar in 1993 30 and Jacobi in 2001 31 Putinar Jacobi representation theorem 1952 John Nash proved that every closed smooth manifold is diffeomorphic to a nonsingular component of a real algebraic set 32 1956 Pierce Birkhoff conjecture formulated 33 Solved in dimensions 2 34 1964 Krivine s Nullstellensatz and Positivestellensatz 35 Rediscovered and popularized by Stengle in 1974 36 Krivine uses real quantifier elimination while Stengle uses Lang s homomorphism theorem 37 1964 Lojasiewicz triangulated semi analytic sets 38 1964 Heisuke Hironaka proved the resolution of singularity theorem 39 1964 Hassler Whitney proved that every analytic variety admits a stratification satisfying the Whitney conditions 40 1967 Theodore Motzkin finds a positive polynomial which is not a sum of squares of polynomials 41 1972 Vladimir Rokhlin proved Gudkov s conjecture 42 1973 Alberto Tognoli proved that every closed smooth manifold is diffeomorphic to a nonsingular real algebraic set 43 1975 George E Collins discovers cylindrical algebraic decomposition algorithm which improves Tarski s real quantifier elimination and allows to implement it on a computer 44 1973 Jean Louis Verdier proved that every subanalytic set admits a stratification with condition w 45 1979 Michel Coste and Marie Francoise Roy discover the real spectrum of a commutative ring 46 1980 Oleg Viro introduced the patch working technique and used it to classify real algebraic curves of low degree 47 Later Ilya Itenberg and Viro used it to produce counterexamples to the Ragsdale conjecture 48 49 and Grigory Mikhalkin applied it to tropical geometry for curve counting 50 1980 Selman Akbulut and Henry C King gave a topological characterization of real algebraic sets with isolated singularities and topologically characterized nonsingular real algebraic sets not necessarily compact 51 1980 Akbulut and King proved that every knot in S n displaystyle S n is the link of a real algebraic set with isolated singularity in R n 1 displaystyle mathbb R n 1 52 1981 Akbulut and King proved that every compact PL manifold is PL homeomorphic to a real algebraic set 53 54 55 1983 Akbulut and King introduced Topological Resolution Towers as topological models of real algebraic sets from this they obtained new topological invariants of real algebraic sets and topologically characterized all 3 dimensional algebraic sets 56 These invariants later generalized by Michel Coste and Krzysztof Kurdyka 57 as well as Clint McCrory and Adam Parusinski 58 1984 Ludwig Brocker s theorem on minimal generation of basic open semialgebraic sets 59 improved and extended to basic closed semialgebraic sets by Scheiderer 60 1984 Benedetti and Dedo proved that not every closed smooth manifold is diffeomorphic to a totally algebraic nonsingular real algebraic set totally algebraic means all its Z 2Z homology cycles are represented by real algebraic subsets 61 1991 Akbulut and King proved that every closed smooth manifold is homeomorphic to a totally algebraic real algebraic set 62 1991 Schmudgen s solution of the multidimensional moment problem for compact semialgebraic sets and related strict positivstellensatz 63 Algebraic proof found by Wormann 64 Implies Reznick s version of Artin s theorem with uniform denominators 65 1992 Akbulut and King proved ambient versions of the Nash Tognoli theorem Every closed smooth submanifold of Rn is isotopic to the nonsingular points component of a real algebraic subset of Rn and they extended this result to immersed submanifolds of Rn 66 67 1992 Benedetti and Marin proved that every compact closed smooth 3 manifold M can be obtained from S 3 displaystyle S 3 by a sequence of blow ups and downs along smooth centers and that M is homeomorphic to a possibly singular affine real algebraic rational threefold 68 1997 Bierstone and Milman proved a canonical resolution of singularities theorem 69 1997 Mikhalkin proved that every closed smooth n manifold can be obtained from S n displaystyle S n by a sequence of topological blow ups and downs 70 1998 Janos Kollar showed that not every closed 3 manifold is a projective real 3 fold which is birational to RP3 71 2000 Scheiderer s local global principle and related non strict extension of Schmudgen s positivstellensatz in dimensions 2 72 73 74 2000 Janos Kollar proved that every closed smooth 3 manifold is the real part of a compact complex manifold which can be obtained from C P 3 displaystyle mathbb CP 3 by a sequence of real blow ups and blow downs 75 2003 Welschinger introduces an invariant for counting real rational curves 76 2005 Akbulut and King showed that not every nonsingular real algebraic subset of RPn is smoothly isotopic to the real part of a nonsingular complex algebraic subset of CPn 77 78 References 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of real algebraic sets with isolated singularities Annals of Mathematics 113 1981 425 446 Selman Akbulut and Henry C King All knots are algebraic Commentarii Mathematici Helvetici 56 Fasc 3 1981 339 351 S Akbulut and H C King Real algebraic structures on topological spaces Publications Mathematiques de l IHES 53 1981 79 162 S Akbulut and L Taylor A topological resolution theorem Publications Mathematiques de l IHES 53 1981 163 196 S Akbulut and H C King The topology of real algebraic sets L Enseignement Mathematique 29 1983 221 261 Selman Akbulut and Henry C King Topology of real algebraic sets MSRI Pub 25 Springer Verlag New York 1992 ISBN 0 387 97744 9 Coste Michel Kurdyka Krzysztof 1992 On the link of a stratum in a real algebraic set Topology 31 2 323 336 doi 10 1016 0040 9383 92 90025 d MR 1167174 McCrory Clint Parusinski Adam 2007 Algebraically constructible functions real algebra and topology Arc spaces and additive invariants in real algebraic and analytic geometry Panoramas et Syntheses vol 24 Paris Societe mathematique de France pp 69 85 arXiv math 0202086 MR 2409689 Brocker Ludwig 1984 Minimale erzeugung von Positivbereichen Geometriae Dedicata in German 16 3 335 350 doi 10 1007 bf00147875 MR 0765338 S2CID 117475206 C Scheiderer Stability index of real varieties Inventiones Mathematicae 97 1989 no 3 467 483 R Benedetti and M Dedo Counterexamples to representing homology classes by real algebraic subvarieties up to homeomorphism Compositio Mathematica 53 1984 143 151 S Akbulut and H C King All compact manifolds are homeomorphic to totally algebraic real algebraic sets Comment Math Helvetici 66 1991 139 149 K Schmudgen The K moment problem for compact semi algebraic sets Math Ann 289 1991 no 2 203 206 T Wormann Strikt Positive Polynome in der Semialgebraischen Geometrie Univ Dortmund 1998 B Reznick Uniform denominators in Hilbert s seventeenth problem Math Z 220 1995 no 1 75 97 S Akbulut and H C King On approximating submanifolds by algebraic sets and a solution to the Nash conjecture Inventiones Mathematicae 107 1992 87 98 S Akbulut and H C King Algebraicity of Immersions Topology vol 31 no 4 1992 701 712 R Benedetti and A Marin Dechirures de varietes de dimension trois Comm Math Helv 67 1992 514 545 E Bierstone and P D Milman Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant Inventiones Mathematicae 128 2 1997 207 302 G Mikhalkin Blow up equivalence of smooth closed manifolds Topology 36 1997 287 299 Janos Kollar The Nash conjecture for algebraic threefolds ERA of AMS 4 1998 63 73 C Scheiderer Sums of squares of regular functions on real algebraic varieties Transactions of the American Mathematical Society 352 2000 no 3 1039 1069 C Scheiderer Sums of squares on real algebraic curves Mathematische Zeitschrift 245 2003 no 4 725 760 C Scheiderer Sums of squares on real algebraic surfaces Manuscripta Mathematica 119 2006 no 4 395 410 Janos Kollar The Nash conjecture for nonprojective threefolds arXiv math 0009108v1 J Y Welschinger Invariants of real rational symplectic 4 manifolds and lower bounds in real enumerative geometry Inventiones Mathematicae 162 2005 no 1 195 234 Zbl 1082 14052 S Akbulut and H C King Transcendental submanifolds of RPn Comm Math Helv 80 2005 427 432 S Akbulut Real algebraic structures Proceedings of GGT 2005 49 58 arXiv math 0601105v3 External links Edit Wikimedia Commons has media related to Real algebraic geometry The Role of Hilbert Problems in Real Algebraic Geometry PostScript Real Algebraic and Analytic Geometry Preprint Server Retrieved from https en wikipedia org w index php title Real algebraic geometry amp oldid 1112649542, wikipedia, wiki, book, books, library,

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