algebraic set
Algebraic may refer to any subject related to algebra in mathematics and related branches like algebraic number theory and algebraic topology. The word algebra itself has several meanings.
Algebraic may also refer to:
* Algebraic data type, a dat ...
algebraic equation
In mathematics, an algebraic equation or polynomial equation is an equation of the form
:P = 0
where ''P'' is a polynomial with coefficients in some field, often the field of the rational numbers. For many authors, the term ''algebraic equation ...
s with real-number coefficients, and mappings between them (in particular real polynomial mappings).
Semialgebraic geometry is the study of
semialgebraic set
In mathematics, a semialgebraic set is a subset ''S'' of ''Rn'' for some real closed field ''R'' (for example ''R'' could be the field of real numbers) defined by a finite sequence of polynomial equations (of the form P(x_1,...,x_n) = 0) and ineq ...
s, i.e. real-number solutions to algebraic
inequalities
Inequality may refer to:
Economics
* Attention inequality, unequal distribution of attention across users, groups of people, issues in etc. in attention economy
* Economic inequality, difference in economic well-being between population groups
* ...
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 In mathematical logic, and more specifically in model theory, an infinite structure (''M'',<,...) which is totally ordered by < is called an o-minimal structure if and only if every real analytic geometry.
Examples:
Real plane curve In mathematics, a real plane curve is usually a real algebraic curve defined in the real projective plane.
Ovals
The field of real numbers is not algebraically closed, the geometry of even a plane curve ''C'' in the real projective plane. Assumin ...
s are examples of real algebraic sets and
polyhedra
In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices.
A convex polyhedron is the convex hull of finitely many points, not all on t ...
are examples of semialgebraic sets. Real
algebraic function In mathematics, an algebraic function is a function that can be defined
as the root of a polynomial equation. Quite often algebraic functions are algebraic expressions using a finite number of terms, involving only the algebraic operations additi ...
s and
Nash function In real algebraic geometry, a Nash function on an open semialgebraic subset ''U'' ⊂ R''n'' is an analytic function
''f'': ''U'' → R satisfying a nontrivial polynomial equation ''P''(''x'',''f''(''x'')) = 0 for all ''x'' in ''U'' (A semialgebra ...
s 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 field
In mathematics, an ordered field is a field together with a total ordering of its elements that is compatible with the field operations. The basic example of an ordered field is the field of real numbers, and every Dedekind-complete ordered fie ...
real closed field
In mathematics, a real closed field is a field ''F'' that has the same first-order properties as the field of real numbers. Some examples are the field of real numbers, the field of real algebraic numbers, and the field of hyperreal numbers. ...
Hilbert's 17th problem
Hilbert's seventeenth problem is one of the 23 Hilbert problems set out in a celebrated list compiled in 1900 by David Hilbert. It concerns the expression of positive definite rational functions as sums of quotients of squares. The original qu ...
and Krivine's Positivestellensatz.) The relation of real algebra to real algebraic geometry is similar to the relation of
commutative algebra
Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Promi ...
to
complex algebraic geometry
In mathematics, complex geometry is the study of geometric structures and constructions arising out of, or described by, the complex numbers. In particular, complex geometry is concerned with the study of spaces such as complex manifolds and ...
convex optimization
Convex optimization is a subfield of mathematical optimization that studies the problem of minimizing convex functions over convex sets (or, equivalently, maximizing concave functions over convex sets). Many classes of convex optimization prob ...
valuation theory In algebra (in particular in algebraic geometry or algebraic number theory), a valuation is a function on a field that provides a measure of size or multiplicity of elements of the field. It generalizes to commutative algebra the notion of size ...
and
model theory
In mathematical logic, model theory is the study of the relationship between formal theories (a collection of sentences in a formal language expressing statements about a mathematical structure), and their models (those structures in which the ...
.
Timeline of real algebra and real algebraic geometry
* 1826 Fourier's algorithm for systems of linear inequalities. Rediscovered by
Lloyd Dines
Lloyd Lyne Dines (29 March 1885, in Shelbyville, Missouri – 17 January 1964, in Quincy, Illinois) was an American-Canadian mathematician, known for his pioneering work on linear inequalities.
in 1919 and
Theodore Motzkin
Theodore Samuel Motzkin (26 March 1908 – 15 December 1970) was an Israeli- American mathematician.
Biography
Motzkin's father Leo Motzkin, a Ukrainian Jew, went to Berlin at the age of thirteen to study mathematics. He pursued university st ...
in 1936.
* 1835
Sturm's theorem
In mathematics, the Sturm sequence of a univariate polynomial is a sequence of polynomials associated with and its derivative by a variant of Euclid's algorithm for polynomials. Sturm's theorem expresses the number of distinct real roots of ...
on real root counting
* 1856 Hermite's theorem on real root counting.
* 1876 Harnack's curve theorem. (This bound on the number of components was later extended to all
Betti number
In algebraic topology, the Betti numbers are used to distinguish topological spaces based on the connectivity of ''n''-dimensional simplicial complexes. For the most reasonable finite-dimensional spaces (such as compact manifolds, finite simplicia ...
s of all real algebraic sets and all semialgebraic sets.)
* 1888 Hilbert's theorem on ternary quartics.
* 1900
Hilbert's problems
Hilbert's problems are 23 problems in mathematics published by German mathematician David Hilbert in 1900. They were all unsolved at the time, and several proved to be very influential for 20th-century mathematics. Hilbert presented ten of the pr ...
17th
17 (seventeen) is the natural number following 16 and preceding 18. It is a prime number.
Seventeen is the sum of the first four prime numbers.
In mathematics
17 is the seventh prime number, which makes seventeen the fourth super-prime, as s ...
problem)
* 1902
Farkas' lemma Farkas' lemma is a solvability theorem for a finite system of linear inequalities in mathematics. It was originally proven by the Hungarian mathematician Gyula Farkas.
Farkas' lemma is the key result underpinning the linear programming duality an ...
(Can be reformulated as linear positivstellensatz.)
*1914
Annibale Comessatti Annibale is the Italian masculine given name and surname equivalent to Hannibal.
In English, it may refer to :
Given name
* Annibale Albani (1682–1751), Italian cardinal
* Annibale I Bentivoglio, (died 1445), ruler of Bologna from 1443
* Annib ...
showed that not every real algebraic surface is birational to RP''2''
* 1916 Fejér's conjecture about nonnegative trigonometric polynomials. (Solved by
Frigyes Riesz
Frigyes Riesz ( hu, Riesz Frigyes, , sometimes spelled as Frederic; 22 January 1880 – 28 February 1956) was a HungarianEberhard Zeidler: Nonlinear Functional Analysis and Its Applications: Linear monotone operators. Springer, 199/ref> mathema ...
.)
* 1927
Emil Artin
Emil Artin (; March 3, 1898 – December 20, 1962) was an Austrian mathematician of Armenian descent.
Artin was one of the leading mathematicians of the twentieth century. He is best known for his work on algebraic number theory, contributing ...
's solution of
Hilbert's 17th problem
Hilbert's seventeenth problem is one of the 23 Hilbert problems set out in a celebrated list compiled in 1900 by David Hilbert. It concerns the expression of positive definite rational functions as sums of quotients of squares. The original qu ...
* 1927 Krull–Baer Theorem (connection between orderings and valuations)
* 1928 Pólya's Theorem on positive polynomials on a simplex
* 1929
B. L. van der Waerden
Bartel Leendert van der Waerden (; 2 February 1903 – 12 January 1996) was a Dutch mathematician and historian of mathematics.
Biography
Education and early career
Van der Waerden learned advanced mathematics at the University of Amst ...
sketches a proof that real algebraic and
semialgebraic set
In mathematics, a semialgebraic set is a subset ''S'' of ''Rn'' for some real closed field ''R'' (for example ''R'' could be the field of real numbers) defined by a finite sequence of polynomial equations (of the form P(x_1,...,x_n) = 0) and ineq ...
s are triangularizable, but the necessary tools have not been developed to make the argument rigorous.
* 1931
Alfred Tarski
Alfred Tarski (, born Alfred Teitelbaum;School of Mathematics and Statistics, University of St Andrews ''School of Mathematics and Statistics, University of St Andrews''. January 14, 1901 – October 26, 1983) was a Polish-American logician ...
Sturm's theorem
In mathematics, the Sturm sequence of a univariate polynomial is a sequence of polynomials associated with and its derivative by a variant of Euclid's algorithm for polynomials. Sturm's theorem expresses the number of distinct real roots of ...
.)
* 1936
Herbert Seifert
Herbert Karl Johannes Seifert (; 27 May 1897, Bernstadt – 1 October 1996, Heidelberg) was a German mathematician known for his work in topology.
Biography
Seifert was born in Bernstadt auf dem Eigen, but soon moved to Bautzen, where he attend ...
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 intersection (from the conclusion of this theorem the word "component" can not be removed).
* 1940 Marshall Stone's representation theorem for partially ordered rings. Improved by
Richard Kadison
Richard Vincent Kadison (July 25, 1925 – August 22, 2018)F ...
in 1951 and Donald Dubois in 1967 (Kadison–Dubois representation theorem). Further improved by Mihai Putinar in 1993 and Jacobi in 2001 (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. (Krivine uses real quantifier elimination while Stengle uses Lang's homomorphism theorem.)
*1964 Lojasiewicz triangulated semi-analytic sets
*1964 Heisuke Hironaka proved the resolution of singularity theorem
*1964
Hassler Whitney
Hassler Whitney (March 23, 1907 – May 10, 1989) was an American mathematician. He was one of the founders of singularity theory, and did foundational work in manifolds, embeddings, immersions, characteristic classes, and geometric integrati ...
proved that every analytic variety admits a stratification satisfying the Whitney conditions.
* 1967
Theodore Motzkin
Theodore Samuel Motzkin (26 March 1908 – 15 December 1970) was an Israeli- American mathematician.
Biography
Motzkin's father Leo Motzkin, a Ukrainian Jew, went to Berlin at the age of thirteen to study mathematics. He pursued university st ...
Alberto Tognoli
Alberto Tognoli (born 26 July 1937, Brescia, died 3 March 2008 in Rapallo) was an Italian mathematician, who worked on algebraic geometry.
Tognoli received his Ph.D. (Laurea) in 1960 from the University of Pisa. From 1970 he became full professor ...
quantifier elimination Quantifier elimination is a concept of simplification used in mathematical logic, model theory, and theoretical computer science. Informally, a quantified statement "\exists x such that \ldots" can be viewed as a question "When is there an x such t ...
and allows to implement it on a computer.
* 1973
Jean-Louis Verdier
Jean-Louis Verdier (; 2 February 1935 – 25 August 1989) was a French mathematician who worked, under the guidance of his doctoral advisor Alexander Grothendieck, on derived categories and Verdier duality. He was a close collaborator of Grothe ...
proved that every subanalytic set admits a stratification with condition (w).
* 1979 Michel Coste and Marie-Françoise Roy discover the real spectrum of a commutative ring.
*1980 Oleg Viro introduced the "patch working" technique and used it to classify real algebraic curves of low degree. Later Ilya Itenberg and Viro used it to produce counterexamples to the
Ragsdale conjecture The Ragsdale conjecture is a mathematical conjecture that concerns the possible arrangements of real algebraic curves embedded in the projective plane. It was proposed by Virginia Ragsdale in her dissertation in 1906 and was disproved in 1979. It h ...
tropical geometry
In mathematics, tropical geometry is the study of polynomials and their geometric properties when addition is replaced with minimization and multiplication is replaced with ordinary addition:
: x \oplus y = \min\,
: x \otimes y = x + y.
So fo ...
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)
*1980 Akbulut and King proved that every knot in is the link of a real algebraic set with isolated singularity in
*1981 Akbulut and King proved that every compact PL manifold is PL homeomorphic to a real algebraic set.
*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. 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 set
In mathematics, a semialgebraic set is a subset ''S'' of ''Rn'' for some real closed field ''R'' (for example ''R'' could be the field of real numbers) defined by a finite sequence of polynomial equations (of the form P(x_1,...,x_n) = 0) and ineq ...
s (improved and extended to basic closed
semialgebraic set
In mathematics, a semialgebraic set is a subset ''S'' of ''Rn'' for some real closed field ''R'' (for example ''R'' could be the field of real numbers) defined by a finite sequence of polynomial equations (of the form P(x_1,...,x_n) = 0) and ineq ...
s by Scheiderer.)
*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).
*1991 Akbulut and King proved that every closed smooth manifold is homeomorphic to a totally algebraic real algebraic set.
* 1991 Schmüdgen's solution of the multidimensional moment problem for compact semialgebraic sets and related strict positivstellensatz. Algebraic proof found by Wörmann. Implies Reznick's version of Artin's theorem with uniform denominators.
*1992 Akbulut and King proved ambient versions of the Nash-Tognoli theorem: Every closed smooth submanifold of R''n'' is isotopic to the nonsingular points (component) of a real algebraic subset of R''n'', and they extended this result to immersed submanifolds of R''n''.
*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
*1997 Bierstone and Milman proved a canonical resolution of singularities theorem
*1997 Mikhalkin proved that every closed smooth n-manifold can be obtained from by a sequence of topological blow ups and downs
*1998 János Kollár showed that not every closed 3-manifold is a projective real 3-fold which is birational to RP''3''
* 2000 Scheiderer's local-global principle and related non-strict extension of Schmüdgen's positivstellensatz in dimensions ≤ 2.
*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.
*2003 Welschinger introduces an invariant for counting real rational curves
*2005 Akbulut and King showed that not every nonsingular real algebraic subset of RP''n'' is smoothly isotopic to the real part of a nonsingular complex algebraic subset of CP''n''S. Akbulut, Real algebraic structures, Proceedings of GGT, (2005) 49–58, arXiv:math/0601105v3.
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) esults 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