Real algebraic geometry
Encyclopedia
In mathematics
, real algebraic geometry is the study of 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 set
s, 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.
.
Examples: Real plane curve
s are examples of real algebraic sets and polyhedra
are examples of semialgebraic sets. Real algebraic function
s 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 field
s) and their applications to the study of positive polynomial
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
to complex algebraic geometry. Related fields are the theory of moment problems, convex optimization, the theory of quadratic forms, valuation theory and model theory
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, real algebraic geometry is the study of real algebraic sets
Algebraic set
In mathematics, an algebraic set over an algebraically closed field K is the set of solutions in Kn of a set of simultaneous equationsand so on up to...
, i.e. real-number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
solutions to algebraic equations with real-number coefficients, and mappings
Map (mathematics)
In most of mathematics and in some related technical fields, the term mapping, usually shortened to map, is either a synonym for function, or denotes a particular kind of function which is important in that branch, or denotes something conceptually similar to a function.In graph theory, a map is a...
between them (in particular real polynomial mappings).
Semialgebraic geometry is the study of semialgebraic set
Semialgebraic set
In mathematics, a semialgebraic set is a subset S of Rn for some real closed field R defined by a finite sequence of polynomial equations and inequalities , or any finite union of such sets. A semialgebraic function is a function with semialgebraic graph...
s, 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: see the Tarski-Seidenberg theorem. Related fields are o-minimal theory and real analytic geometrySubanalytic set
In mathematics, particularly in the subfield of real analytic geometry, a subanalytic set is a set of points defined in a way broader than for semianalytic sets...
.
Examples: Real plane curve
Real plane curve
In mathematics, a real plane curve is usually a real algebraic curve defined in the real projective plane.-Ovals:Since the real number field is not algebraically closed, the geometry of even a plane curve C in the real projective plane is not a very easy topic...
s are examples of real algebraic sets and polyhedra
Polyhedron
In elementary geometry a polyhedron is a geometric solid in three dimensions with flat faces and straight edges...
are examples of semialgebraic sets. Real algebraic function
Algebraic function
In mathematics, an algebraic function is informally a function that satisfies a polynomial equation whose coefficients are themselves polynomials with rational coefficients. For example, an algebraic function in one variable x is a solution y for an equationwhere the coefficients ai are polynomial...
s and Nash functions are examples of semialgebraic mappings. Piecewise polynomial mappings (see the Pierce-Birkhoff conjecture
Pierce–Birkhoff conjecture
In abstract algebra, the Pierce–Birkhoff conjecture asserts that any piecewise-polynomial function can be expressed as a maximum of finite minima of finite collections of polynomials. It was first stated, albeit in non-rigorous and vague wording, in the 1956 paper of Garrett Birkhoff and Richard S....
) 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
Cylindrical algebraic decomposition
Given a set of polynomials in Rn and a set S in Rn the Cylindrical algebraic decomposition algorithm finds a decomposition of S in to a number of cells such that for each cell each polynomial has constant sign.-References:...
. 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
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. Historically, the axiomatization of an ordered field was abstracted gradually from the real numbers, by mathematicians including David Hilbert, Otto Hölder and...
and ordered rings
Ordered ring
In abstract algebra, an ordered ring is a 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 0 ≤ ab....
(in particular real closed field
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.-Definitions:...
s) and their applications to the study of positive polynomial
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 space ℝn...
s and sums-of-squares of polynomials
Polynomial SOS
In mathematics, a form h of degree 2m in the real n-dimensional vector x is sum of squares of forms if and only if there exist forms g_1,\ldots,g_k of degree m such that...
. (See Hilbert's 17th problem
Hilbert's seventeenth 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 entails expression of definite rational functions as quotients of sums of squares...
and Krivine's Positivestellensatz
Stengle's Positivstellensatz
In real semialgebraic geometry, Stengle's Positivstellensatz characterizes polynomials that are positive on a semialgebraic set, which is defined by systems of inequalities of polynomials with real coefficients, or more generally, coefficients from any real-closed field.It can be thought of as an...
.) The relation of real algebra to real algebraic geometry is similar to the relation of commutative algebra
Commutative algebra
Commutative algebra is the branch of abstract algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on 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
Model theory
In mathematics, model theory is the study of mathematical structures using tools from mathematical logic....
.
Timeline of real algebra and real algebraic geometry
- 1826 Fourier's algorithmFourier–Motzkin eliminationFourier–Motzkin elimination, FME method, is a mathematical algorithm for eliminating variables from a system of linear inequalities. It can look for both real and integer solutions...
for systems of linear inequalities. Rediscovered by Dines in 1919 and Motzkin in 1936 - 1835 Sturm's theoremSturm's theoremIn mathematics, Sturm's theorem is a symbolic procedure to determine the number of distinct real roots of a polynomial. It was named for Jacques Charles François Sturm...
on real root counting - 1856 Hermite's theorem on real root counting
- 1876 Harnack's curve theoremHarnack's curve theoremIn real algebraic geometry, Harnack's curve theorem, named after Axel Harnack, describes the possible numbers of connected components that an algebraic curve can have, in terms of the degree of the curve...
(This bound on the number of components was later extended to all Betti numbers of all real algebraic sets and all semialgebraic sets ) - 1888 Hilbert's theorem on ternary quartics.
- 1900 Hilbert's problemsHilbert's problemsHilbert's problems form a list of twenty-three problems in mathematics published by German mathematician David Hilbert in 1900. The problems were all unsolved at the time, and several of them were very influential for 20th century mathematics...
(especially the 16thHilbert's sixteenth problemHilbert's 16th problem was posed by David Hilbert at the Paris conference of the International Congress of Mathematicians in 1900, together with the other 22 problems....
and the 17thHilbert's seventeenth problemHilbert's seventeenth problem is one of the 23 Hilbert problems set out in a celebrated list compiled in 1900 by David Hilbert. It entails expression of definite rational functions as quotients of sums of squares...
problem) - 1902 Farkas lemma (Can be reformulated as linear positivstellensatz.)
- 1916 Fejer's conjecture about nonnegative trigonometric polynomials. (Solved by Riesz. )
- 1927 Artin's solution of Hilbert's 17th problem
- 1927 Krull-Baer Theorem (connection between orderings and valuations)
- 1928 Polya's Theorem on positive polynomials on a simplex
- 1929 van der Waerden proves that real algebraic and semialgebraic setSemialgebraic setIn mathematics, a semialgebraic set is a subset S of Rn for some real closed field R defined by a finite sequence of polynomial equations and inequalities , or any finite union of such sets. A semialgebraic function is a function with semialgebraic graph...
s are triangularizable. - 1931 Tarski's real quantifier eliminationTarski–Seidenberg theoremIn mathematics, the Tarski–Seidenberg theorem states that a set in -dimensional space defined by polynomial identities and inequalities can be projected down onto n-dimensional space, and the resulting set is still definable in terms of polynomial identities and inequalities...
. Improved and popularized by Seidenberg in 1954. (Both use Sturm's theoremSturm's theoremIn mathematics, Sturm's theorem is a symbolic procedure to determine the number of distinct real roots of a polynomial. It was named for Jacques Charles François Sturm...
.) - 1940 Stone's representation theorem for partially ordered rings. Improved by Kadison in 1951 and Dubois in 1967 (Kadison-Dubois representation theorem). Further improved by Putinar in 1993 and Jacobi in 2001 (Putinar-Jacobi representation theorem).
- 1956 Pierce–Birkhoff conjecturePierce–Birkhoff conjectureIn abstract algebra, the Pierce–Birkhoff conjecture asserts that any piecewise-polynomial function can be expressed as a maximum of finite minima of finite collections of polynomials. It was first stated, albeit in non-rigorous and vague wording, in the 1956 paper of Garrett Birkhoff and Richard S....
formulated. (Solved in dimensions ≤ 2.) - 1964 Krivine's Nullstellensatz and PositivestellensatzStengle's PositivstellensatzIn real semialgebraic geometry, Stengle's Positivstellensatz characterizes polynomials that are positive on a semialgebraic set, which is defined by systems of inequalities of polynomials with real coefficients, or more generally, coefficients from any real-closed field.It can be thought of as an...
.. Rediscovered and popularized by Stengle in 1974 (Krivine uses real quantifier eliminationTarski–Seidenberg theoremIn mathematics, the Tarski–Seidenberg theorem states that a set in -dimensional space defined by polynomial identities and inequalities can be projected down onto n-dimensional space, and the resulting set is still definable in terms of polynomial identities and inequalities...
while Stengle uses Lang's homomorphism theorem. ) - 1967 Motzkin finds a positive polynomial which is not a sum of squares of polynomialsPolynomial SOSIn mathematics, a form h of degree 2m in the real n-dimensional vector x is sum of squares of forms if and only if there exist forms g_1,\ldots,g_k of degree m such that...
. - 1973 Tognoli proves that every compact smooth manifold is diffeomorphic to a real algebraic set. (Topological characterization of real algebraic sets with singularities is known only in small dimensions.)
- 1975 Collins discovers cylindrical algebraic decompositionCylindrical algebraic decompositionGiven a set of polynomials in Rn and a set S in Rn the Cylindrical algebraic decomposition algorithm finds a decomposition of S in to a number of cells such that for each cell each polynomial has constant sign.-References:...
algorithm. - 1979 Coste and Roy discover the real spectrum of a commutative ring.
- 1984 Bröcker's theorem on minimal generation of basic open semialgebraic setSemialgebraic setIn mathematics, a semialgebraic set is a subset S of Rn for some real closed field R defined by a finite sequence of polynomial equations and inequalities , or any finite union of such sets. A semialgebraic function is a function with semialgebraic graph...
s(Improved and extended to basic closed semialgebraic setSemialgebraic setIn mathematics, a semialgebraic set is a subset S of Rn for some real closed field R defined by a finite sequence of polynomial equations and inequalities , or any finite union of such sets. A semialgebraic function is a function with semialgebraic graph...
s by Scheiderer) - 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.
- 2000 Scheiderer's local-global principle and related non-strict extension of Schmüdgen's positivstellensatz in dimensions ≤ 2.