Real algebraic geometry

In mathematics, real algebraic geometry is the study of 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 Dines in 1919^[4] and 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 Comessatti showed that not every real algebraic surface is birational to RP^2[15]
• 1916 Fejér's conjecture about nonnegative trigonometric polynomials.^[16] (Solved by Riesz.^[17])
• 1927 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 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 Tarski's real quantifier elimination.^[23] Improved and popularized by Seidenberg in 1954.^[24] (Both use Sturm's theorem.)
• 1936 Seifert proved that every closed smooth submanifold of Rⁿ with trivial normal bundle, can be isotoped to a component of a nonsingular real algebraic subset of Rⁿ which is a complete intersection^[25] (from the conclusion of this theorem the word "component" can not be removed^[26]).
• 1940 Stone's representation theorem for partially ordered rings.^[27] Improved by Kadison in 1951^[28] and Dubois in 1967^[29] (Kadison–Dubois representation theorem). Further improved by Putinar in 1993^[30] and Jacobi in 2001^[31] (Putinar–Jacobi representation theorem).
• 1952 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 Hironaka proved the resolution of singularity theorem^[39]
• 1964 Whitney proved that every analytic variety admits a stratification satisfying the Whitney conditions.^[40]
• 1967 Motzkin finds a positive polynomial which is not a sum of squares of polynomials.^[41]
• 1973 Tognoli proved that every closed smooth manifold is diffeomorphic to a nonsingular real algebraic set.^[42]
• 1975 Collins discovers cylindrical algebraic decomposition algorithm, which improves Tarski's real quantifier elimination and allows to implement it on a computer.^[43]
• 1973 Verdier proved that every subanalytic set admits a stratification with condition (w).^[44]
• 1979 Coste and Roy discover the real spectrum of a commutative ring.^[45]
• 1980 Viro introduced the "patch working" technique and used it to classify real algebraic curves of low degree.^[46] Later Itenberg-Viro used it to produce counterexamples to the Ragsdale conjecture,^[47][48] and Mikhalkin applied it to "tropical geometry" for curve counting.^[49]
• 1980 Akbulut and King gave a topological characterization of real algebraic sets with isolated singularities, and topologically characterized nonsingular real algebraic sets (not necessarily compact)^[50]
• 1980 Akbulut and King proved that every knot in Sⁿ is the link of a real algebraic set with isolated singularity in R^n+1[51]
• 1981 Akbulut and King proved that every compact PL manifold is PL homeomorphic to a real algebraic set.^[52][53][54]
• 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.^[55] These invariants later generalized by Coste-Kurdyka^[56] and McCrory-Parusinski^[57]
• 1984 Bröcker's theorem on minimal generation of basic open semialgebraic sets^[58](Improved and extended to basic closed semialgebraic sets by Scheiderer^[59])
• 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).^[60]
• 1991 Akbulut and King proved that every closed smooth manifold is homeomorphic to a totally algebraic real algebraic set.^[61]
• 1991 Schmüdgen's solution of the multidimensional moment problem for compact semialgebraic sets and related strict positivstellensatz.^[62] Algebraic proof found by Wörmann.^[63] Implies Reznick's version of Artin's theorem with uniform denominators.^[64]
• 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.^[65][66]
• 1992 Benedetti and Marin proved that every compact closed smooth 3-manifold M can be obtained from S³ 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^[67]
• 1997 Bierstone and Milman proved a canonical resolution of singularities theorem^[68]
• 1997 Mikhalkin proved that every closed smooth n-manifold can be obtained from Sⁿ by a sequence of topological blow ups and downs^[69]
• 1998 Kollar showed that not every closed 3-manifold is a projective real 3-fold which is birational to RP^3[70]
• 2000 Scheiderer's local-global principle and related non-strict extension of Schmüdgen's positivstellensatz in dimensions ≤ 2.^[71][72][73]
• 2000 Kollar proved that every closed smooth 3–manifold is the real part of a compact complex manifold which can be obtained from CP³ by a sequence of real blow ups and downs^[74]
• 2003 Welschinger introduces an invariant for counting real rational curves^[75]
• 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 CP^n[76][77]

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. 248. Cambridge University Press. p. 31. Zbl 0953.03045.
2. Khovanskii, A. G. (1991). Fewnomials. Translations of Mathematical Monographs. 88. Translated from the Russian by Smilka Zdravkovska. Providence, RI: American Mathematical Society. ISBN 0-8218-4547-0. Zbl 0728.12002.
3. J. B. J. Fourier, Solution d'une question particuliére du calcul des inégalités. Bull. sci. Soc. Philomn. Paris 99–100. OEuvres 2, 315–319.
4. L. L. Dines, Systems of linear inequalities. Ann. of Math. (2) 20 (1919), no. 3, 191–199.
5. Motzkin, T. Beiträge zur Theorie der linearen Ungleichungen. IV+ 76 S. Diss., Basel (1936).
6. J. C. F. Sturm, Mémoires divers présentés par des savants étrangers 6, pp. 273–318 (1835).
7. C. Hermite, Sur le Nombre des Racines d’une Équation Algébrique Comprise Entre des Limites Données, J. Reine Angew. Math., vol. 52, pp. 39–51 (1856).
8. C. G. A. Harnack Über Vieltheiligkeit der ebenen algebraischen Curven, Math. Ann. 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. J. Milnor, On the Betti numbers of real varieties, Proc. Amer. Math. Soc. 15 (1964), 275–280.
11. R. 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. S. Basu, On bounding the Betti numbers and computing the Euler characteristic of semi-algebraic sets, Discrete Comput. Geom. 22 (1999), no. 1, 1–18.
13. D. Hilbert, ¨Uber die Darstellung definiter Formen als Summe von Formenquadraten. Math. Ann. 32, 342–350 (1888).
14. J. Farkas, "Über die Theorie der Einfachen Ungleichungen", Journal für die Reine und Angewandte Mathematik 124, 1–27
15. A. Comessatti, Sulla connessione delle superfizie razionali reali, An- nali di Math. 23(3) (1914) 215-283.
16. L. Fej´er, ¨Uber trigonometrische Polynome, J. Reine Angew. Math. 146 (1916), 53–82.
17. F. Riesz and B. Sz.-Nagy, Functional Analysis, Frederick Ungar Publ. Co., New York, 1955.
18. E. Artin, Uber die Zerlegung definiter Funktionen in Quadrate, Abh. Math. Sem. Univ. Hamburg, 5 (1927), 85–99.
19. W. Krull, Allgemeine Bewertungstheorie. J. Reine Angew. Math. 167, 160–196 (1932).
20. R. Baer, Über nicht-archimedisch geordnete Körper. (Beiträge zur Algebra 5.). Sitzungsberichte Heidelberg 1927, 8. Abh., 3–13 (1927).
21. G. 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. van der Waerden, B. L. Topologische Begründung des Kalküls der abzählenden Geometrie. Math. Ann. 102, 337–362 (1929).
23. A. 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. A. Seidenberg, A new decision method for elementary algebra, Ann. of Math. 60 (1954), 365–374.
25. H. Seifert, Algebraische approximation von Mannigfaltigkeiten, Math. Zeitschrift, 41 (1936), 1-17
26. S. Akbulut and H.C. King, Submanifolds and homology of nonsingular real algebraic varieties, American Jour of Math, vol. 107, no. 1 (Feb., 1985) p.72
27. M. H. Stone, A general theory of spectra. I. Proc. Natl. Acad. Sci. U.S.A. 26, (1940). 280–283.
28. R. V. Kadison, A representation theory for commutative topological algebra. Mem. Am. Math. Soc. 7, 39 p. (1951).
29. Dubois, D. W. A note on David Harrison's theory of preprimes. Pacific J. Math. 21 1967 15–19.
30. M. Putinar, Positive polynomials on compact semi-algebraic sets. Indiana Univ. Math. J. 42 (1993), no. 3, 969–984.
31. T. Jacobi, A representation theorem for certain partially ordered commutative rings. Math. Z. 237 (2001), no. 2, 259–273.
32. J. Nash, Real algebraic manifolds, Ann. of Math. 56 (1952), 405–421
33. G. Birkhoff, R.S. Pierce, Lattice ordered rings, An. Acad. Brasil. Ciˆenc. 28 (1956) 41–69.
34. L. Mah´e, On the Pierce–Birkhoff conjecture, Ordered fields and real algebraic geometry (Boulder, Colo., 1983). Rocky Mountain J. Math. 14 (1984), no. 4, 983–985.
35. J.-L. Krivine, Anneaux préordonnés, J. Analyse Math. 12 (1964), 307–326.
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. H. Hironaka , Resolution of singularities of an algebraic variety over a field of characteristic zero. I, Ann. of Math. (2) 79 (1): (1964) 109–203, and part II, pp. 205–326.
40. H. Whitney, Local properties of analytic varieties, Differential and combinatorial topology (ed. S. Cairns), Princeton Univ. Press, Princeton N.J. (1965), 205--244.
41. T. S. Motzkin, The arithmetic-geometric inequality. 1967 Inequalities (Proc. Sympos. Wright–Patterson Air Force Base, Ohio, 1965) pp. 205–224.
42. A. Tognoli, Su una congettura di Nash. Ann. Sc. Norm. Super. Pisa 27, 167–185 (1973).
43. G. E. Collins, "Quantifier Elimination for the Elementary Theory of Real Closed Fields by Cylindrical Algebraic Decomposition." Lect. Notes Comput. Sci. 33, 134–183, 1975.
44. J.-L. Verdier, Stratifications de Whitney et théorème de Bertini-Sard, Inventiones Math. 36, 295-312 (1976).
45. M.-F. Coste-Roy, M. Coste, Topologies for real algebraic geometry. Topos theoretic methods in geometry, pp. 37–100, Various Publ. Ser., 30, Aarhus Univ., Aarhus, 1979.
46. O. 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 Math., pages187–200. Springer, Berlin, 1984
47. 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 Soviet Mathematics - Doklady. 22: 566–570. 1980. Zbl 0422.14032. Missing or empty |title= (help)
48. Itenberg, Ilia; Mikhalkin, Grigory; Shustin, Eugenii (2007). Tropical algebraic geometry. Oberwolfach Seminars. 35. Basel: Birkhäuser. pp. 34–35. ISBN 978-3-7643-8309-1. Zbl 1162.14300.
49. Mikhalkin, G.: Enumerative tropical algebraic geometry in R2. J. Amer. Math. Soc. 18 (2005), 313–377
50. S. Akbulut and H.C. King, The topology of real algebraic sets with isolated singularities, Ann. of Math. 113 (1981), 425–446.
51. S. Akbulut and H.C. King, All knots are algebraic, Comm. Math. Helv. 56, Fasc. 3 (1981), 339–351.
52. S. Akbulut and H.C. King, Real algebraic structures on topological spaces, Pub I.H.E.S. 53 (1981), 79–162.
53. S. Akbulut and L. Taylor, A topological resolution theorem, Pub. I.H.E.S., 53 (1981), 163–196.
54. S. Akbulut and H.C. King, The topology of real algebraic sets, L'Enseignement Math. 29 (1983), 221-261.
55. S. Akbulut and H.C. King, Topology of real algebraic sets, MSRI Pub, 25. Springer-Verlag, New York (1992) ISBN 0-387-97744-9
56. M.Coste and K. Kurdyka, On the link of a stratum in a real algebraic set, Topology 31 (1992) 323-336
57. C. McCrory and A. Parusinski, Algebraically constructible functions: real algebra and topology, arXiv:math/0202086v1.
58. L. Bröcker, Minimale Erzeugung von Positivbereichen. Geom. Dedicata 16, 335–350 (1984).
59. C. Scheiderer, Stability index of real varieties. Invent. Math. 97 (1989), no. 3, 467–483.
60. R. Benedetti and M. Dedo, Counterexamples to representing homology classes by real algebraic subvarieties up to homeomorphism, Compositio Mathematica, 53, (1984), 143-151.
61. S. Akbulut and H.C. King, All compact manifolds are homeomorphic to totally algebraic real algebraic sets, Comment. Math. Helvetici 66 (1991) 139-149.
62. K. Schmüdgen, The K-moment problem for compact semi-algebraic sets. Math. Ann. 289 (1991), no. 2, 203-206.
63. T. Wörmann Strikt Positive Polynome in der Semialgebraischen Geometrie, Univ. Dortmund 1998.
64. B. Reznick, Uniform denominators in Hilbert's seventeenth problem. Math. Z. 220 (1995), no. 1, 75–97.
65. S. Akbulut and H.C. King On approximating submanifolds by algebraic sets and a solution to the Nash conjecture, Invent.Math.107 (1992), 87–98
66. S. Akbulut and H.C. King, Algebraicity of Immersions, Topology, vol. 31, no. 4, (1992), 701–712.
67. R. Benedetti and A. Marin , Dechirures de varietes de dimension trois ...., Comm. Math. Helv. 67 (1992), 514–545.
68. E. Bierstone and P.D. Milman , Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant, Invent. Math. 128 (2) (1997) 207–302.
69. G. Mikhalkin, Blow up equivalence of smooth closed manifolds, Topology, 36 (1997) 287–299
70. J. Kollar, The Nash conjecture for algebraic threefolds, ERA of AMS 4 (1998) 63–73
71. C. Scheiderer, Sums of squares of regular functions on real algebraic varieties. Trans. Amer. Math. Soc. 352 (2000), no. 3, 1039–1069.
72. C. Scheiderer, Sums of squares on real algebraic curves. Math. Z. 245 (2003), no. 4, 725–760.
73. C. Scheiderer, Sums of squares on real algebraic surfaces. Manuscripta Math. 119 (2006), no. 4, 395–410.
74. J. Kollar, The Nash conjecture for nonprojective threefolds, arXiv:math/0009108v1
75. J.-Y. Welschinger, Invariants of real rational symplectic 4-manifolds and lower bounds in real enumerative geometry, Invent. Math. 162 (2005), no. 1, 195-234. Zbl 1082.14052
76. S. Akbulut and H.C. King, Transcendental submanifolds of RPⁿ Comm. Math. Helv., 80, (2005), 427–432
77. S. Akbulut, Real algebraic structures, Proceedings of GGT, (2005) 49–58, arXiv:math/0601105v3.

External links

• The Role of Hilbert Problems in Real Algebraic Geometry (PostScript)
• Real Algebraic and Analytic Geometry Preprint Server