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. 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 the Krivine-Stengle Positivstellensatz.) 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.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 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.) 1888 Hilbert's theorem on ternary quartics. 1900 Hilbert's problems (especially the 16th and the 17th problem) 1902 Farkas' lemma (Can be reformulated as linear positivstellensatz.) 1914 Annibale Comessatti showed that not every real algebraic surface is birational to RP2 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 1927 Krull–Baer Theorem (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 with trivial normal bundle, can be isotoped to a component of a nonsingular real algebraic subset of 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. Improved by Richard Kadison in 1951 and Donald Dubois in 1967 (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. 1956 Pierce–Birkhoff conjecture formulated. (Solved in dimensions ≤ 2.) 1964 Krivine's Nullstellensatz and Positivestellensatz. 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 . 1972 Vladimir Rokhlin proved Gudkov's conjecture."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 . 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, Translated in and Grigory Mikhalkin applied it to tropical geometry for curve counting. 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 is the link of a real algebraic set with isolated singularity in 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) These invariants later generalized by Michel Coste and Krzysztof Kurdyka as well as Clint McCrory and Adam Parusiński. 1984 Ludwig Bröcker's theorem on minimal generation of basic open semialgebraic sets (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.)