In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, real algebraic geometry is the sub-branch of
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
studying real
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 ...
s, i.e.
real-number solutions to
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 mapping
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* Real (L'Arc-en-Ciel album), ''Real'' (L'Arc-en-Ciel album) (2000)
* Real ...
s).
Semialgebraic geometry is the study of
semialgebraic set
In mathematics, a semialgebraic set is a subset ''S'' of ''Rn'' for some real closed field ''R'' (for example ''R'' could be the field of real numbers) defined by a finite sequence of polynomial equations (of the form P(x_1,...,x_n) = 0) and inequa ...
s, i.e. real-number solutions to algebraic
inequalities
Inequality may refer to:
Economics
* Attention inequality, unequal distribution of attention across users, groups of people, issues in etc. in attention economy
* Economic inequality, difference in economic well-being between population groups
* ...
with-real number coefficients, and mappings between them. The most natural mappings between semialgebraic sets are
semialgebraic mappings, i.e., mappings whose graphs are semialgebraic sets.
Terminology
Nowadays the words 'semialgebraic geometry' and 'real algebraic geometry' are used as synonyms, because real algebraic sets cannot be studied seriously without the use of semialgebraic sets. For example, a projection of a real algebraic set along a coordinate axis need not be a real algebraic set, but it is always a semialgebraic set: this is the
Tarski–Seidenberg theorem In mathematics, the Tarski–Seidenberg theorem states that a set in (''n'' + 1)-dimensional space defined by polynomial equations and inequalities can be projected down onto ''n''-dimensional space, and the resulting set is still defina ...
. Related fields are
o-minimal theory In mathematical logic, and more specifically in model theory, an infinite structure (''M'',<,...) which is totally ordered by < is called an o-minimal structure if and only if every and
real analytic geometry
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010)
...
.
Examples:
Real plane curves 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 additio ...
s and
Nash functions are examples of semialgebraic mappings. Piecewise polynomial mappings (see the
Pierce–Birkhoff conjecture In abstract algebra, the Pierce–Birkhoff conjecture asserts that any piecewise-polynomial function can be expressed as a Supremum, maximum of finite Infimum, minima of finite collections of polynomials. It was first stated, albeit in non-Mathemati ...
) 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
In mathematics, cylindrical algebraic decomposition (CAD) is a notion, and an algorithm to compute it, that are fundamental for computer algebra and real algebraic geometry. Given a set ''S'' of polynomials in R''n'', a cylindrical algebraic decom ...
. 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 field ...
s and
ordered ring
In abstract algebra, an ordered ring is a (usually commutative) ring ''R'' with a total order ≤ such that for all ''a'', ''b'', and ''c'' in ''R'':
* if ''a'' ≤ ''b'' then ''a'' + ''c'' ≤ ''b'' + ''c''.
* if 0 ≤ ''a'' and 0 ≤ ''b'' then ...
s (in particular
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.
Def ...
s) and their applications to the study of
positive polynomial
In mathematics, a positive polynomial on a particular set is a polynomial whose values are positive on that set.
Let ''p'' be a polynomial in ''n'' variables with real coefficients and let ''S'' be a subset of the ''n''-dimensional Euclidean ...
s 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
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. Prominent ...
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 c ...
. Related fields are the theory of
moment problem
In mathematics, a moment problem arises as the result of trying to invert the mapping that takes a measure ''μ'' to the sequences of moments
:m_n = \int_^\infty x^n \,d\mu(x)\,.
More generally, one may consider
:m_n = \int_^\infty M_n(x) ...
s,
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 probl ...
, the theory of
quadratic forms
In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example,
:4x^2 + 2xy - 3y^2
is a quadratic form in the variables and . The coefficients usually belong to ...
,
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 inhe ...
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 s ...
.
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 studi ...
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 loca ...
on real root counting
* 1856 Hermite's theorem on real root counting.
* 1876
Harnack's curve theorem
In real algebraic geometry, Harnack's curve theorem, named after Axel Harnack, gives the possible numbers of connected components that an algebraic curve can have, in terms of the degree of the curve. For any algebraic curve of degree in the ...
. (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 pro ...
(especially the
16th
16 (sixteen) is the natural number following 15 and preceding 17. 16 is a composite number, and a square number, being 42 = 4 × 4. It is the smallest number with exactly five divisors, its proper divisors being , , and .
In English speech, ...
and the
17th 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 (natural scientist), Gyula Farkas.
Farkas' Lemma (mathematics), lemma is the key ...
(Can be reformulated as linear positivstellensatz.)
*1914
Annibale Comessatti 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 lar ...
'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 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 Amsterd ...
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 inequa ...
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 a ...
's
real quantifier elimination. Improved and popularized by
Abraham Seidenberg
Abraham Seidenberg (June 2, 1916 – May 3, 1988) was an American mathematician.
Early life
Seidenberg was born on June 2, 1916 to Harry and Fannie Seidenberg in Washington D.C. He graduated with a B.A. from the University of Maryland in 1937. ...
in 1954. (Both use
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 loca ...
.)
* 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
Marshall Harvey Stone (April 8, 1903 – January 9, 1989) was an American mathematician who contributed to real analysis, functional analysis, topology and the study of Boolean algebras.
Biography
Stone was the son of Harlan Fiske Stone, who wa ...
's representation theorem for partially ordered rings. Improved by
Richard Kadison
Richard Vincent Kadison (July 25, 1925 – August 22, 2018)[F ...](_blank)
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 In abstract algebra, the Pierce–Birkhoff conjecture asserts that any piecewise-polynomial function can be expressed as a Supremum, maximum of finite Infimum, minima of finite collections of polynomials. It was first stated, albeit in non-Mathemati ...
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
is a Japanese mathematician who was awarded the Fields Medal in 1970 for his contributions to algebraic geometry.
Career
Hironaka entered Kyoto University in 1949. After completing his undergraduate studies at Kyoto University, he received his ...
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 integration t ...
proved that every analytic variety admits a stratification satisfying the
Whitney conditions In differential topology, a branch of mathematics, the Whitney conditions are conditions on a pair of submanifolds of a manifold introduced by Hassler Whitney in 1965.
A stratification of a topological space is a finite filtration by closed subsets ...
.
* 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 studi ...
finds a positive polynomial which is not a
sum of squares of polynomials.
* 1972
Vladimir Rokhlin proved
Gudkov's conjecture In real algebraic geometry, Gudkov's conjecture, also called Gudkov’s congruence, (named after Dmitry Gudkov) was a conjecture, and is now a theorem, which states that an M-curve of even degree 2d obeys the congruence
: p - n \equiv d^2\, (\! ...
.
* 1973
Alberto Tognoli proved that every closed smooth manifold is diffeomorphic to a nonsingular real algebraic set.
* 1975
George E. Collins
George E. Collins (January 10, 1928 in Stuart, Iowa – November 21, 2017 in Madison, Wisconsin) was an American mathematician and computer scientist. He is the inventor of Garbage collection (computer science), garbage collection by reference co ...
discovers
cylindrical algebraic decomposition
In mathematics, cylindrical algebraic decomposition (CAD) is a notion, and an algorithm to compute it, that are fundamental for computer algebra and real algebraic geometry. Given a set ''S'' of polynomials in R''n'', a cylindrical algebraic decom ...
algorithm, which improves Tarski's real
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
Marie-Françoise Roy (born 28 April 1950 in Paris) is a French mathematician noted for her work in real algebraic geometry. She has been Professor of Mathematics at the University of Rennes 1 since 1985 and in 2009 was made a ''Chevalier'' of the ...
discover the real spectrum of a commutative ring.
*1980
Oleg Viro
Oleg Yanovich Viro (russian: Олег Янович Виро) (b. 13 May 1948, Leningrad, USSR) is a Russian mathematician in the fields of topology and algebraic geometry, most notably real algebraic geometry, tropical geometry and knot theory.
...
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, and
Grigory Mikhalkin Grigory, Grigori and Grigoriy are Russian language, Russian masculine given names.
It may refer to watcher angels or more specifically to Watcher (angel)#Grigori, the egrḗgoroi or Watcher angels.
Grigory
* Grigory Baklanov (1923–2009), Russ ...
applied it to
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 f ...
for curve counting.
*1980
Selman Akbulut
Selman Akbulut (born 1949) is a Turkish mathematician, specializing in research in topology, and geometry. He was a professor at Michigan State University until February 2020.
Career
In 1975 he earned his Ph.D. from the University of California ...
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 inequa ...
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 inequa ...
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
János Kollár (born 7 June 1956) is a Hungarian mathematician, specializing in algebraic geometry.
Professional career
Kollár began his studies at the Eötvös University in Budapest and later received his PhD at Brandeis University in 1984 ...
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
János Kollár (born 7 June 1956) is a Hungarian mathematician, specializing in algebraic geometry.
Professional career
Kollár began his studies at the Eötvös University in Budapest and later received his PhD at Brandeis University in 1984 ...
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
Notes
External links
{{Commonscat
''The Role of Hilbert Problems in Real Algebraic Geometry'' (PostScript)''Real Algebraic and Analytic Geometry Preprint Server''