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

\Rn

with trivial normal bundle, can be isotoped to a component of a nonsingular real algebraic subset of

\Rn

which is a complete intersection[24] (from the conclusion of this theorem the word "component" can not be removed[25]).

Sn

is the link of a real algebraic set with isolated singularity in

\Rn+1

[50]

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[65]

Sn

by a sequence of topological blow ups and downs[67]

CP3

by a sequence of real blow ups and blow downs.[72]

References

External links

Notes and References

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    . Askold Khovanskii . Fewnomials . Translated from the Russian by Smilka Zdravkovska . 0728.12002 . Translations of Mathematical Monographs . 88 . Providence, RI . . 1991 . 0-8218-4547-0 .

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