Real algebraic geometry

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.{{cite book | first=L. | last=van den Dries | title=Tame topology and o-minimal structures | series=London Mathematical Society Lecture Note Series | volume=248 | publisher=Cambridge University Press | year=1998 | zbl=0953.03045 | page=31 }}{{cite book | last=Khovanskii | first=A. G. | authorlink= Askold Khovanskii | title=Fewnomials | others= Translated from the Russian by Smilka Zdravkovska | zbl=0728.12002 | series=Translations of Mathematical Monographs | volume=88 | location=Providence, RI | publisher=American Mathematical Society | year=1991 | isbn=0-8218-4547-0 }} 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.

• 1826 Fourier's algorithm for systems of linear inequalities.Joseph B. J. Fourier, Solution d'une question particuliére du calcul des inégalités. Bull. sci. Soc. Philomn. Paris 99–100. OEuvres 2, 315–319. Rediscovered by Lloyd Dines in 1919{{cite journal|authorlink=Lloyd Dines|first=Lloyd L. |last=Dines|title= Systems of linear inequalities|journal= Annals of Mathematics |series=(2) |volume= 20 |year=1919|issue=3|pages=191–199|doi=10.2307/1967869 |jstor=1967869 }} and Theodore Motzkin in 1936.Theodore Motzkin, Beiträge zur Theorie der linearen Ungleichungen. IV+ 76 S. Diss., Basel (1936).
• 1835 Sturm's theorem on real root countingJacques Charles François Sturm, Mémoires divers présentés par des savants étrangers 6, pp. 273–318 (1835).
• 1856 Hermite's theorem on real root counting.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).
• 1876 Harnack's curve theorem.C. G. A. Harnack Über Vieltheiligkeit der ebenen algebraischen Curven, Mathematische Annalen 10 (1876), 189–199 (This bound on the number of components was later extended to all Betti numbers of all real algebraic setsI. G. Petrovski˘ı and O. A. Ole˘ınik, On the topology of real algebraic surfaces, Izvestiya Akad. Nauk SSSR. Ser.Mat. 13, (1949). 389–402John Milnor, On the Betti numbers of real varieties, Proceedings of the American Mathematical Society 15 (1964), 275–280.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. and all semialgebraic sets.{{cite journal|first=Saugata|last= Basu|title= On bounding the Betti numbers and computing the Euler characteristic of semi-algebraic sets|journal= Discrete & Computational Geometry |volume=22 |year=1999|issue= 1|pages= 1–18|doi= 10.1007/PL00009443|s2cid= 7023328|hdl= 2027.42/42421|hdl-access= free}})
• 1888 Hilbert's theorem on ternary quartics.{{cite journal|first=David|last=Hilbert|authorlink=David Hilbert|title=Uber die Darstellung definiter Formen als Summe von Formenquadraten|journal=Mathematische Annalen|volume= 32|pages= 342–350 |year=1888|issue=3 |doi=10.1007/BF01443605 |s2cid=177804714 |url=https://zenodo.org/record/1428214 }}
• 1900 Hilbert's problems (especially the 16th and the 17th problem)
• 1902 Farkas' lemma{{cite journal|authorlink=Gyula Farkas (natural scientist)|first=Julius|last= Farkas|url=http://dz-srv1.sub.uni-goettingen.de/sub/digbib/loader?ht=VIEW&did=D261364|title=Über die Theorie der Einfachen Ungleichungen|journal= Journal für die Reine und Angewandte Mathematik|volume= 124|pages= 1–27}} (Can be reformulated as linear positivstellensatz.)
• 1914 Annibale Comessatti showed that not every real algebraic surface is birational to RP²{{cite journal|first=Annibale |last=Comessatti|title= Sulla connessione delle superfizie razionali reali| journal=Annali di Matematica Pura ed Applicata |volume=23|issue=3|year=1914|pages= 215–283|doi=10.1007/BF02419577 |s2cid=121297483 |url=https://zenodo.org/record/1947509 }}
• 1916 Fejér's conjecture about nonnegative trigonometric polynomials.Lipót Fejér, ¨Uber trigonometrische Polynome, J. Reine Angew. Math. 146 (1916), 53–82. (Solved by Frigyes Riesz.Frigyes Riesz and Béla Szőkefalvi-Nagy, Functional Analysis, Frederick Ungar Publ. Co., New York, 1955.)
• 1927 Emil Artin's solution of Hilbert's 17th problem{{cite journal|first=Emil|last= Artin|authorlink=Emil Artin|title= Uber die Zerlegung definiter Funktionen in Quadrate|journal= Abh. Math. Sem. Univ. Hamburg|volume= 5 |year=1927|pages= 85–99|doi= 10.1007/BF02952512|s2cid= 122881707}}
• 1927 Krull–Baer Theorem{{cite journal| first=Wolfgang|last=Krull| authorlink=Wolfgang Krull|title= Allgemeine Bewertungstheorie|journal= Journal für die reine und angewandte Mathematik|volume= 1932|pages= 160–196 |year=1932|issue=167 |doi=10.1515/crll.1932.167.160 |s2cid=199547002 }}{{citation|first=Reinhold| last=Baer| authorlink=Reinhold Baer| title= Über nicht-archimedisch geordnete Körper |journal= Sitzungsberichte der Heidelberger Akademie der Wissenschaften. Mathematisch-Naturwissenschaftliche Klasse|volume= 8|pages= 3–13 |year=1927}} (connection between orderings and valuations)
• 1928 Pólya's Theorem on positive polynomials on a simplexGeorge 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
• 1929 B. L. van der Waerden sketches a proof that real algebraic and semialgebraic sets are triangularizable,B. L. van der Waerden, Topologische Begründung des Kalküls der abzählenden Geometrie. Math. Ann. 102, 337–362 (1929). but the necessary tools have not been developed to make the argument rigorous.
• 1931 Alfred Tarski's real quantifier elimination.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. Improved and popularized by Abraham Seidenberg in 1954.Abraham Seidenberg, A new decision method for elementary algebra, Annals of Mathematics 60 (1954), 365–374. (Both use Sturm's theorem.)
• 1936 Herbert Seifert proved that every closed smooth submanifold of $\backslash R^n$ with trivial normal bundle, can be isotoped to a component of a nonsingular real algebraic subset of $\backslash R^n$ which is a complete intersectionHerbert Seifert, Algebraische approximation von Mannigfaltigkeiten, Mathematische Zeitschrift 41 (1936), 1–17 (from the conclusion of this theorem the word "component" can not be removedSelman 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).
• 1940 Marshall Stone's representation theorem for partially ordered rings.{{cite journal|first=Marshall|last=Stone|authorlink=Marshall Stone|title= A general theory of spectra. I. |journal= Proceedings of the National Academy of Sciences of the United States of America |volume= 26|year=1940|issue=4 |pages= 280–283|doi=10.1073/pnas.26.4.280 |pmid=16588355 |pmc=1078172 |doi-access=free |bibcode=1940PNAS...26..280S }} Improved by Richard Kadison in 1951{{citation|authorlink=Richard Kadison|first=Richard V. |last=Kadison|title= A representation theory for commutative topological algebra|journal= Memoirs of the American Mathematical Society |volume=7| pages=39 pp |year=1951|mr=0044040}} and Donald Dubois in 1967{{cite journal|last=Dubois|first= Donald W. |title=A note on David Harrison's theory of preprimes|journal= Pacific Journal of Mathematics |volume=21 |year=1967|pages= 15–19|doi= 10.2140/pjm.1967.21.15 |mr=0209200|s2cid= 120262803 |url=https://projecteuclid.org/euclid.pjm/1102992597|doi-access=free}} (Kadison–Dubois representation theorem). Further improved by Mihai Putinar in 1993Mihai Putinar, Positive polynomials on compact semi-algebraic sets. Indiana University Mathematics Journal 42 (1993), no. 3, 969–984. and Jacobi in 2001T. Jacobi, A representation theorem for certain partially ordered commutative rings. Mathematische Zeitschrift 237 (2001), no. 2, 259–273. (Putinar–Jacobi representation theorem).
• 1952 John Nash proved that every closed smooth manifold is diffeomorphic to a nonsingular component of a real algebraic set.{{cite journal|authorlink=John Forbes Nash Jr.|first=John|last= Nash | title= Real algebraic manifolds|journal=Annals of Mathematics |volume=56 |year=1952|issue=3 |pages= 405–421|doi=10.2307/1969649 |jstor=1969649 }}
• 1956 Pierce–Birkhoff conjecture formulated.{{cite journal|first1=Garrett|last1= Birkhoff|author1-link= Garrett Birkhoff |first2=Richard Scott|last2= Pierce|title= Lattice ordered rings|journal= Anais da Academia Brasileira de Ciências|volume= 28 |year=1956|pages= 41–69}} (Solved in dimensions ≤ 2.{{cite journal|first=Louis|last= Mahé|title= On the Pierce–Birkhoff conjecture| journal=Rocky Mountain Journal of Mathematics|volume= 14 |year=1984| issue= 4|pages= 983–985|mr=0773148|doi=10.1216/RMJ-1984-14-4-983|doi-access=free}})
• 1964 Krivine's Nullstellensatz and Positivestellensatz.{{cite journal | last1=Krivine | first1=J.-L. | url=http://hal.archives-ouvertes.fr/docs/00/16/56/58/PDF/Anneaux_preordonnes.pdf | title=Anneaux préordonnés | journal= Journal d'Analyse Mathématique | volume=12 | date=1964 | pages=307–326 | doi=10.1007/BF02807438 | doi-access=free}} Rediscovered and popularized by Stengle in 1974.G. Stengle, A nullstellensatz and a positivstellensatz in semialgebraic geometry. Math. Ann. 207 (1974), 87–97. (Krivine uses real quantifier elimination while Stengle uses Lang's homomorphism theorem.S. Lang, Algebra. Addison–Wesley Publishing Co., Inc., Reading, Mass. 1965 xvii+508 pp.)
• 1964 Lojasiewicz triangulated semi-analytic setsS. Lojasiewicz, Triangulation of semi-analytic sets, Ann. Scu. Norm. di Pisa, 18 (1964), 449–474.
• 1964 Heisuke Hironaka proved the resolution of singularity theoremHeisuke 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.
• 1964 Hassler Whitney proved that every analytic variety admits a stratification satisfying the Whitney conditions.Hassler Whitney, Local properties of analytic varieties, Differential and combinatorial topology (ed. S. Cairns), Princeton Univ. Press, Princeton N.J. (1965), 205–244.
• 1967 Theodore Motzkin finds a positive polynomial which is not a sum of squares of polynomials.Theodore S. Motzkin, The arithmetic-geometric inequality. 1967 Inequalities (Proc. Sympos. Wright-Patterson Air Force Base, Ohio, 1965) pp. 205–224 {{MR|0223521}}.
• 1972 Vladimir Rokhlin proved Gudkov's conjecture."[https://www.maths.ed.ac.uk/~v1ranick/papers/rohlin3.pdf Proof of Gudkov's hypothesis]". V. A. Rokhlin. Functional Analysis and Its Applications, volume 6, pp. 136–138 (1972)
• 1973 Alberto Tognoli proved that every closed smooth manifold is diffeomorphic to a nonsingular real algebraic set.Alberto Tognoli, Su una congettura di Nash, Annali della Scuola Normale Superiore di Pisa 27, 167–185 (1973).
• 1975 George E. Collins discovers cylindrical algebraic decomposition algorithm, which improves Tarski's real quantifier elimination and allows to implement it on a computer.George E. Collins, "Quantifier elimination for real closed fields by cylindrical algebraic decomposition", Lect. Notes Comput. Sci. 33, 134–183, 1975 {{MR|0403962}}.
• 1973 Jean-Louis Verdier proved that every subanalytic set admits a stratification with condition (w).Jean-Louis Verdier, Stratifications de Whitney et théorème de Bertini-Sard, Inventiones Mathematicae 36, 295–312 (1976).
• 1979 Michel Coste and Marie-Françoise Roy discover the real spectrum of a commutative ring.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.
• 1980 Oleg Viro introduced the "patch working" technique and used it to classify real algebraic curves of low degree.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 Later Ilya Itenberg and Viro used it to produce counterexamples to the Ragsdale conjecture,{{cite journal| last=Viro | first=Oleg Ya. | authorlink=Oleg Viro | year=1980 | title=Кривые степени 7, кривые степени 8 и гипотеза Рэгсдейл |trans-title=Curves of degree 7, curves of degree 8 and the hypothesis of Ragsdale | journal=Doklady Akademii Nauk SSSR | volume=254 | issue=6 | pages=1306–1309}} Translated in {{cite journal| title=Curves of degree 7, curves of degree 8 and Ragsdale's conjecture | journal=Soviet Mathematics - Doklady | volume=22 | pages=566–570 | year=1980 | zbl=0422.14032 }}{{cite book | last1=Itenberg | first1=Ilia | last2=Mikhalkin | first2=Grigory | last3=Shustin | first3=Eugenii | title=Tropical algebraic geometry | zbl=1162.14300 | series=Oberwolfach Seminars | volume=35 | location=Basel | publisher=Birkhäuser | isbn=978-3-7643-8309-1 | year=2007 | pages=34–35 }} and Grigory Mikhalkin applied it to tropical geometry for curve counting.{{cite journal |first=Grigory|last= Mikhalkin|title= Enumerative tropical algebraic geometry in $\backslash R^2$|journal= Journal of the American Mathematical Society |volume=18 |year=2005|pages= 313–377|doi= 10.1090/S0894-0347-05-00477-7|doi-access= free}}
• 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)Selman Akbulut and Henry C. King, The topology of real algebraic sets with isolated singularities, Annals of Mathematics 113 (1981), 425–446.
• 1980 Akbulut and King proved that every knot in $S^n$ is the link of a real algebraic set with isolated singularity in $\backslash R^\{n+1\}$Selman Akbulut and Henry C. King, All knots are algebraic, Commentarii Mathematici Helvetici 56, Fasc. 3 (1981), 339–351.
• 1981 Akbulut and King proved that every compact PL manifold is PL homeomorphic to a real algebraic set.S. Akbulut and H.C. King, Real algebraic structures on topological spaces,

Publications Mathématiques de l'IHÉS 53 (1981), 79–162.S. Akbulut and L. Taylor, A topological resolution theorem,

Publications Mathématiques de l'IHÉS 53 (1981), 163–196.S. Akbulut and H.C. King, The topology of real algebraic sets,

L'Enseignement Mathématique 29 (1983), 221–261.

• 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.Selman Akbulut and Henry C. King, Topology of real algebraic sets, MSRI Pub, 25. Springer-Verlag, New York (1992) {{ISBN|0-387-97744-9}} These invariants later generalized by Michel Coste and Krzysztof Kurdyka

{{cite journal | last1=Coste | first1=Michel | last2=Kurdyka | first2=Krzysztof | title=On the link of a stratum in a real algebraic set | journal=Topology | volume=31 | issue=2 | year=1992 | doi=10.1016/0040-9383(92)90025-d | pages=323–336 | mr=1167174 | doi-access=free }} as well as Clint McCrory and Adam Parusiński.{{citation|first1=Clint|last1= McCrory|first2= Adam|last2= Parusiński|contribution= Algebraically constructible functions: real algebra and topology| arxiv=math/0202086|title=Arc spaces and additive invariants in real algebraic and analytic geometry|pages= 69–85|

series=Panoramas et Synthèses|volume= 24|publisher= Société mathématique de France |location= Paris|year= 2007|mr=2409689}}

• 1984 Ludwig Bröcker's theorem on minimal generation of basic open semialgebraic sets{{cite journal | last=Bröcker | first=Ludwig | title=Minimale erzeugung von Positivbereichen | journal=Geometriae Dedicata | volume=16 | issue=3 | year=1984 | pages=335–350| doi=10.1007/bf00147875 | language=de | mr=0765338| s2cid=117475206 }}

(improved and extended to basic closed semialgebraic sets by Scheiderer.C. Scheiderer, Stability index of real varieties. Inventiones Mathematicae 97 (1989), no. 3, 467–483.)

• 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).R. Benedetti and M. Dedo, Counterexamples to representing homology classes by real algebraic subvarieties up to homeomorphism, Compositio Mathematica, 53, (1984), 143–151.
• 1991 Akbulut and King proved that every closed smooth manifold is homeomorphic to a totally algebraic real algebraic set.S. Akbulut and H.C. King, All compact manifolds are homeomorphic to totally algebraic real algebraic sets, Comment. Math. Helv. 66 (1991) 139–149.
• 1991 Schmüdgen's solution of the multidimensional moment problem for compact semialgebraic sets and related strict positivstellensatz.K. Schmüdgen, The K-moment problem for compact semi-algebraic sets. Math. Ann. 289 (1991), no. 2, 203–206. Algebraic proof found by Wörmann.T. Wörmann Strikt Positive Polynome in der Semialgebraischen Geometrie, Univ. Dortmund 1998. Implies Reznick's version of Artin's theorem with uniform denominators.B. Reznick, Uniform denominators in Hilbert's seventeenth problem. Math. Z. 220 (1995), no. 1, 75–97.
• 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.S. Akbulut and H.C. King On approximating submanifolds by algebraic sets and a solution to the Nash conjecture, Inventiones Mathematicae 107 (1992), 87–98S. Akbulut and H.C. King, Algebraicity of Immersions, Topology, vol. 31, no. 4, (1992), 701–712.
• 1992 Benedetti and Marin proved that every compact closed smooth 3-manifold M can be obtained from $S^3$ by a sequence of blow ups and downs along smooth centers, and that M is homeomorphic to a possibly singular affine real algebraic rational threefoldR. Benedetti and A. Marin, Déchirures de variétés de dimension trois ...., Comment. Math. Helv. 67 (1992), 514–545.
• 1997 Bierstone and Milman proved a canonical resolution of singularities theoremE. 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.
• 1997 Mikhalkin proved that every closed smooth n-manifold can be obtained from $S^n$ by a sequence of topological blow ups and downsG. Mikhalkin, Blow up equivalence of smooth closed manifolds, Topology, 36 (1997) 287–299
• 1998 János Kollár showed that not every closed 3-manifold is a projective real 3-fold which is birational to RP³János Kollár, The Nash conjecture for algebraic threefolds, ERA of AMS 4 (1998) 63–73
• 2000 Scheiderer's local-global principle and related non-strict extension of Schmüdgen's positivstellensatz in dimensions ≤ 2.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.
• 2000 János Kollár proved that every closed smooth 3–manifold is the real part of a compact complex manifold which can be obtained from $\backslash mathbb\{CP\}^3$ by a sequence of real blow ups and blow downs.János Kollár, The Nash conjecture for nonprojective threefolds, arXiv:math/0009108v1
• 2003 Welschinger introduces an invariant for counting real rational curvesJ.-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}}
• 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 CPⁿS. Akbulut and H.C. King, Transcendental submanifolds of RPⁿ Comment. Math. Helv., 80, (2005), 427–432S. Akbulut, Real algebraic structures, Proceedings of GGT, (2005) 49–58, arXiv:math/0601105v3.

• 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

{{Reflist}}

{{Commons category}}

• [http://www.mathematik.uni-bielefeld.de/~kersten/hilbert/prob17.ps The Role of Hilbert Problems in Real Algebraic Geometry (PostScript)]
• [http://www.maths.manchester.ac.uk/raag/ Real Algebraic and Analytic Geometry Preprint Server]