Real algebraic geometry explained
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
with trivial normal bundle, can be isotoped to a component of a nonsingular real algebraic subset of
which is a complete intersection
[24] (from the conclusion of this theorem the word "component" can not be removed
[25]).
is the link of a real algebraic set with isolated singularity in
[50] - 1981 Akbulut and King proved that every compact PL manifold is PL homeomorphic to a real algebraic set.[51] [52] [53]
- 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.[54] These invariants later generalized by Michel Coste and Krzysztof Kurdyka[55] as well as Clint McCrory and Adam Parusiński.
- 1984 Ludwig Bröcker's theorem on minimal generation of basic open semialgebraic sets[56] (improved and extended to basic closed semialgebraic sets by Scheiderer.[57])
- 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).[58]
- 1991 Akbulut and King proved that every closed smooth manifold is homeomorphic to a totally algebraic real algebraic set.[59]
- 1991 Schmüdgen's solution of the multidimensional moment problem for compact semialgebraic sets and related strict positivstellensatz.[60] Algebraic proof found by Wörmann.[61] Implies Reznick's version of Artin's theorem with uniform denominators.[62]
- 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.[63] [64]
- 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
[65] - 1997 Bierstone and Milman proved a canonical resolution of singularities theorem[66]
- 1997 Mikhalkin proved that every closed smooth n-manifold can be obtained from
by a sequence of topological blow ups and downs
[67] - 1998 János Kollár showed that not every closed 3-manifold is a projective real 3-fold which is birational to RP3[68]
- 2000 Scheiderer's local-global principle and related non-strict extension of Schmüdgen's positivstellensatz in dimensions ≤ 2.[69] [70] [71]
- 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.
[72] - 2003 Welschinger introduces an invariant for counting real rational curves[73]
- 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[74] [75]
References
- S. Akbulut and H.C. King, Topology of real algebraic sets, MSRI Pub, 25. Springer-Verlag, New York (1992)
- 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.
- 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. ; 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. ; 0-8218-4402-4
External links
Notes and References
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