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Algebraic geometry is a branch of
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 ...
, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of
abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The ter ...
ic techniques, mainly from
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. Prom ...
, for solving geometrical problems about these sets of zeros. The fundamental objects of study in algebraic geometry are algebraic varieties, which are geometric manifestations of solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are: plane algebraic curves, which include lines,
circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is cons ...
s,
parabola In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves. One descri ...
s, ellipses,
hyperbola In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, ca ...
s, cubic curves like
elliptic curve In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. I ...
s, and quartic curves like
lemniscate In algebraic geometry, a lemniscate is any of several figure-eight or -shaped curves. The word comes from the Latin "''lēmniscātus''" meaning "decorated with ribbons", from the Greek λημνίσκος meaning "ribbons",. or which alternative ...
s and Cassini ovals. A point of the plane belongs to an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of the points of special interest like the
singular point Singularity or singular point may refer to: Science, technology, and mathematics Mathematics * Mathematical singularity, a point at which a given mathematical object is not defined or not "well-behaved", for example infinite or not differentiab ...
s, the
inflection point In differential calculus and differential geometry, an inflection point, point of inflection, flex, or inflection (British English: inflexion) is a point on a smooth plane curve at which the curvature changes sign. In particular, in the case ...
s and the points at infinity. More advanced questions involve the
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
of the curve and relations between the curves given by different equations. Algebraic geometry occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex analysis,
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
and
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Ma ...
. Initially a study of systems of polynomial equations in several variables, the subject of algebraic geometry starts where equation solving leaves off, and it becomes even more important to understand the intrinsic properties of the totality of solutions of a system of equations, than to find a specific solution; this leads into some of the deepest areas in all of mathematics, both conceptually and in terms of technique. In the 20th century, algebraic geometry split into several subareas. * The mainstream of algebraic geometry is devoted to the study of the complex points of the algebraic varieties and more generally to the points with coordinates in an
algebraically closed field In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in . Examples As an example, the field of real numbers is not algebraically closed, because ...
. * Real algebraic geometry is the study of the real points of an algebraic variety. * Diophantine geometry and, more generally, arithmetic geometry is the study of the points of an algebraic variety with coordinates in fields that are not algebraically closed and occur in algebraic number theory, such as the field of
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
s, number fields,
finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...
s, function fields, and ''p''-adic fields. * A large part of singularity theory is devoted to the singularities of algebraic varieties. * Computational algebraic geometry is an area that has emerged at the intersection of algebraic geometry and
computer algebra In mathematics and computer science, computer algebra, also called symbolic computation or algebraic computation, is a scientific area that refers to the study and development of algorithms and software for manipulating mathematical expressions ...
, with the rise of computers. It consists mainly of
algorithm In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing ...
design and
software Software is a set of computer programs and associated documentation and data. This is in contrast to hardware, from which the system is built and which actually performs the work. At the lowest programming level, executable code consist ...
development for the study of properties of explicitly given algebraic varieties. Much of the development of the mainstream of algebraic geometry in the 20th century occurred within an abstract algebraic framework, with increasing emphasis being placed on "intrinsic" properties of algebraic varieties not dependent on any particular way of embedding the variety in an ambient coordinate space; this parallels developments in
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
, differential and complex geometry. One key achievement of this abstract algebraic geometry is Grothendieck's scheme theory which allows one to use sheaf theory to study algebraic varieties in a way which is very similar to its use in the study of differential and analytic manifolds. This is obtained by extending the notion of point: In classical algebraic geometry, a point of an affine variety may be identified, through Hilbert's Nullstellensatz, with a maximal ideal of the coordinate ring, while the points of the corresponding affine scheme are all prime ideals of this ring. This means that a point of such a scheme may be either a usual point or a subvariety. This approach also enables a unification of the language and the tools of classical algebraic geometry, mainly concerned with complex points, and of algebraic number theory. Wiles' proof of the longstanding conjecture called Fermat's Last Theorem is an example of the power of this approach.


Basic notions


Zeros of simultaneous polynomials

In classical algebraic geometry, the main objects of interest are the vanishing sets of collections of
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exampl ...
s, meaning the set of all points that simultaneously satisfy one or more polynomial equations. For instance, the two-dimensional
sphere A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the c ...
of radius 1 in three-dimensional
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
R3 could be defined as the set of all points (''x'',''y'',''z'') with :x^2+y^2+z^2-1=0.\, A "slanted" circle in R3 can be defined as the set of all points (''x'',''y'',''z'') which satisfy the two polynomial equations :x^2+y^2+z^2-1=0,\, :x+y+z=0.\,


Affine varieties

First we start with a field ''k''. In classical algebraic geometry, this field was always the complex numbers C, but many of the same results are true if we assume only that ''k'' is algebraically closed. We consider the affine space of dimension ''n'' over ''k'', denoted An(''k'') (or more simply A''n'', when ''k'' is clear from the context). When one fixes a coordinate system, one may identify An(''k'') with ''k''''n''. The purpose of not working with ''k''''n'' is to emphasize that one "forgets" the vector space structure that ''k''n carries. A function ''f'' : A''n'' → A1 is said to be ''polynomial'' (or ''regular'') if it can be written as a polynomial, that is, if there is a polynomial ''p'' in ''k'' 'x''1,...,''x''''n''such that ''f''(''M'') = ''p''(''t''1,...,''t''''n'') for every point ''M'' with coordinates (''t''1,...,''t''''n'') in A''n''. The property of a function to be polynomial (or regular) does not depend on the choice of a coordinate system in A''n''. When a coordinate system is chosen, the regular functions on the affine ''n''-space may be identified with the ring of polynomial functions in ''n'' variables over ''k''. Therefore, the set of the regular functions on A''n'' is a ring, which is denoted ''k'' ''A''n'' We say that a polynomial ''vanishes'' at a point if evaluating it at that point gives zero. Let ''S'' be a set of polynomials in ''k'' ''An The ''vanishing set of S'' (or ''vanishing locus'' or ''zero set'') is the set ''V''(''S'') of all points in A''n'' where every polynomial in ''S'' vanishes. Symbolically, :V(S) = \.\, A subset of A''n'' which is ''V''(''S''), for some ''S'', is called an ''algebraic set''. The ''V'' stands for ''variety'' (a specific type of algebraic set to be defined below). Given a subset ''U'' of A''n'', can one recover the set of polynomials which generate it? If ''U'' is ''any'' subset of A''n'', define ''I''(''U'') to be the set of all polynomials whose vanishing set contains ''U''. The ''I'' stands for
ideal Ideal may refer to: Philosophy * Ideal (ethics), values that one actively pursues as goals * Platonic ideal, a philosophical idea of trueness of form, associated with Plato Mathematics * Ideal (ring theory), special subsets of a ring considered ...
: if two polynomials ''f'' and ''g'' both vanish on ''U'', then ''f''+''g'' vanishes on ''U'', and if ''h'' is any polynomial, then ''hf'' vanishes on ''U'', so ''I''(''U'') is always an ideal of the polynomial ring ''k'' ''A''n'' Two natural questions to ask are: * Given a subset ''U'' of A''n'', when is ''U'' = ''V''(''I''(''U''))? * Given a set ''S'' of polynomials, when is ''S'' = ''I''(''V''(''S''))? The answer to the first question is provided by introducing the Zariski topology, a topology on A''n'' whose closed sets are the algebraic sets, and which directly reflects the algebraic structure of ''k'' ''A''n'' Then ''U'' = ''V''(''I''(''U'')) if and only if ''U'' is an algebraic set or equivalently a Zariski-closed set. The answer to the second question is given by Hilbert's Nullstellensatz. In one of its forms, it says that ''I''(''V''(''S'')) is the radical of the ideal generated by ''S''. In more abstract language, there is a Galois connection, giving rise to two
closure operator In mathematics, a closure operator on a set ''S'' is a function \operatorname: \mathcal(S)\rightarrow \mathcal(S) from the power set of ''S'' to itself that satisfies the following conditions for all sets X,Y\subseteq S : Closure operators are de ...
s; they can be identified, and naturally play a basic role in the theory; the example is elaborated at Galois connection. For various reasons we may not always want to work with the entire ideal corresponding to an algebraic set ''U''. Hilbert's basis theorem implies that ideals in ''k'' ''A''n''are always finitely generated. An algebraic set is called '' irreducible'' if it cannot be written as the union of two smaller algebraic sets. Any algebraic set is a finite union of irreducible algebraic sets and this decomposition is unique. Thus its elements are called the ''irreducible components'' of the algebraic set. An irreducible algebraic set is also called a '' variety''. It turns out that an algebraic set is a variety if and only if it may be defined as the vanishing set of a prime ideal of the polynomial ring. Some authors do not make a clear distinction between algebraic sets and varieties and use ''irreducible variety'' to make the distinction when needed.


Regular functions

Just as
continuous function In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in val ...
s are the natural maps on
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...
s and smooth functions are the natural maps on differentiable manifolds, there is a natural class of functions on an algebraic set, called ''regular functions'' or ''polynomial functions''. A regular function on an algebraic set ''V'' contained in A''n'' is the restriction to ''V'' of a regular function on A''n''. For an algebraic set defined on the field of the complex numbers, the regular functions are smooth and even
analytic Generally speaking, analytic (from el, ἀναλυτικός, ''analytikos'') refers to the "having the ability to analyze" or "division into elements or principles". Analytic or analytical can also have the following meanings: Chemistry * ...
. It may seem unnaturally restrictive to require that a regular function always extend to the ambient space, but it is very similar to the situation in a normal
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...
, where the Tietze extension theorem guarantees that a continuous function on a closed subset always extends to the ambient topological space. Just as with the regular functions on affine space, the regular functions on ''V'' form a ring, which we denote by ''k'' 'V'' This ring is called the '' coordinate ring of V''. Since regular functions on V come from regular functions on A''n'', there is a relationship between the coordinate rings. Specifically, if a regular function on ''V'' is the restriction of two functions ''f'' and ''g'' in ''k'' ''A''n'' then ''f'' − ''g'' is a polynomial function which is null on ''V'' and thus belongs to ''I''(''V''). Thus ''k'' 'V''may be identified with ''k'' ''A''n''''I''(''V'').


Morphism of affine varieties

Using regular functions from an affine variety to A1, we can define regular maps from one affine variety to another. First we will define a regular map from a variety into affine space: Let ''V'' be a variety contained in A''n''. Choose ''m'' regular functions on ''V'', and call them ''f''1, ..., ''f''''m''. We define a ''regular map'' ''f'' from ''V'' to A''m'' by letting . In other words, each ''f''''i'' determines one coordinate of the range of ''f''. If ''V''′ is a variety contained in A''m'', we say that ''f'' is a ''regular map'' from ''V'' to ''V''′ if the range of ''f'' is contained in ''V''′. The definition of the regular maps apply also to algebraic sets. The regular maps are also called ''morphisms'', as they make the collection of all affine algebraic sets into a category, where the objects are the affine algebraic sets and the morphisms are the regular maps. The affine varieties is a subcategory of the category of the algebraic sets. Given a regular map ''g'' from ''V'' to ''V''′ and a regular function ''f'' of ''k'' 'V''′ then . The map is a ring homomorphism from ''k'' 'V''′to ''k'' 'V'' Conversely, every ring homomorphism from ''k'' 'V''′to ''k'' 'V''defines a regular map from ''V'' to ''V''′. This defines an
equivalence of categories In category theory, a branch of abstract mathematics, an equivalence of categories is a relation between two categories that establishes that these categories are "essentially the same". There are numerous examples of categorical equivalences fr ...
between the category of algebraic sets and the opposite category of the finitely generated reduced ''k''-algebras. This equivalence is one of the starting points of scheme theory.


Rational function and birational equivalence

In contrast to the preceding sections, this section concerns only varieties and not algebraic sets. On the other hand, the definitions extend naturally to projective varieties (next section), as an affine variety and its projective completion have the same field of functions. If ''V'' is an affine variety, its coordinate ring is an integral domain and has thus a field of fractions which is denoted ''k''(''V'') and called the ''field of the rational functions'' on ''V'' or, shortly, the '' function field'' of ''V''. Its elements are the restrictions to ''V'' of the
rational function In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be ...
s over the affine space containing ''V''. The
domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined ** Domain of definition of a partial function ** Natural domain of a partial function **Domain of holomorphy of a function * ...
of a rational function ''f'' is not ''V'' but the complement of the subvariety (a hypersurface) where the denominator of ''f'' vanishes. As with regular maps, one may define a ''rational map'' from a variety ''V'' to a variety ''V'''. As with the regular maps, the rational maps from ''V'' to ''V''' may be identified to the
field homomorphism Field theory is the branch of mathematics in which fields are studied. This is a glossary of some terms of the subject. (See field theory (physics) for the unrelated field theories in physics.) Definition of a field A field is a commutative rin ...
s from ''k''(''V''') to ''k''(''V''). Two affine varieties are ''birationally equivalent'' if there are two rational functions between them which are
inverse Inverse or invert may refer to: Science and mathematics * Inverse (logic), a type of conditional sentence which is an immediate inference made from another conditional sentence * Additive inverse (negation), the inverse of a number that, when a ...
one to the other in the regions where both are defined. Equivalently, they are birationally equivalent if their function fields are isomorphic. An affine variety is a ''
rational variety In mathematics, a rational variety is an algebraic variety, over a given field ''K'', which is birationally equivalent to a projective space of some dimension over ''K''. This means that its function field is isomorphic to :K(U_1, \dots , U_d), th ...
'' if it is birationally equivalent to an affine space. This means that the variety admits a ''rational parameterization'', that is a parametrization with
rational function In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be ...
s. For example, the circle of equation x^2+y^2-1=0 is a rational curve, as it has the parametric equation :x=\frac :y=\frac\,, which may also be viewed as a rational map from the line to the circle. The problem of resolution of singularities is to know if every algebraic variety is birationally equivalent to a variety whose projective completion is nonsingular (see also
smooth completion In algebraic geometry, the smooth completion (or smooth compactification) of a smooth affine algebraic curve ''X'' is a complete smooth algebraic curve which contains ''X'' as an open subset. Smooth completions exist and are unique over a perfect ...
). It was solved in the affirmative in characteristic 0 by Heisuke Hironaka in 1964 and is yet unsolved in finite characteristic.


Projective variety

Just as the formulas for the roots of second, third, and fourth degree polynomials suggest extending real numbers to the more algebraically complete setting of the complex numbers, many properties of algebraic varieties suggest extending affine space to a more geometrically complete projective space. Whereas the complex numbers are obtained by adding the number ''i'', a root of the polynomial , projective space is obtained by adding in appropriate points "at infinity", points where parallel lines may meet. To see how this might come about, consider the variety . If we draw it, we get a
parabola In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves. One descri ...
. As ''x'' goes to positive infinity, the slope of the line from the origin to the point (''x'', ''x''2) also goes to positive infinity. As ''x'' goes to negative infinity, the slope of the same line goes to negative infinity. Compare this to the variety ''V''(''y'' − ''x''3). This is a cubic curve. As ''x'' goes to positive infinity, the slope of the line from the origin to the point (''x'', ''x''3) goes to positive infinity just as before. But unlike before, as ''x'' goes to negative infinity, the slope of the same line goes to positive infinity as well; the exact opposite of the parabola. So the behavior "at infinity" of ''V''(''y'' − ''x''3) is different from the behavior "at infinity" of ''V''(''y'' − ''x''2). The consideration of the ''projective completion'' of the two curves, which is their prolongation "at infinity" in the projective plane, allows us to quantify this difference: the point at infinity of the parabola is a regular point, whose tangent is the line at infinity, while the point at infinity of the cubic curve is a cusp. Also, both curves are rational, as they are parameterized by ''x'', and the Riemann-Roch theorem implies that the cubic curve must have a singularity, which must be at infinity, as all its points in the affine space are regular. Thus many of the properties of algebraic varieties, including birational equivalence and all the topological properties, depend on the behavior "at infinity" and so it is natural to study the varieties in projective space. Furthermore, the introduction of projective techniques made many theorems in algebraic geometry simpler and sharper: For example, Bézout's theorem on the number of intersection points between two varieties can be stated in its sharpest form only in projective space. For these reasons, projective space plays a fundamental role in algebraic geometry. Nowadays, the '' projective space'' P''n'' of dimension ''n'' is usually defined as the set of the lines passing through a point, considered as the origin, in the affine space of dimension , or equivalently to the set of the vector lines in a vector space of dimension . When a coordinate system has been chosen in the space of dimension , all the points of a line have the same set of coordinates, up to the multiplication by an element of ''k''. This defines the
homogeneous coordinates In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work , are a system of coordinates used in projective geometry, just as Cartesian coordinates are used in Euclidean geometr ...
of a point of P''n'' as a sequence of elements of the base field ''k'', defined up to the multiplication by a nonzero element of ''k'' (the same for the whole sequence). A polynomial in variables vanishes at all points of a line passing through the origin if and only if it is homogeneous. In this case, one says that the polynomial ''vanishes'' at the corresponding point of P''n''. This allows us to define a ''projective algebraic set'' in P''n'' as the set , where a finite set of homogeneous polynomials vanishes. Like for affine algebraic sets, there is a bijection between the projective algebraic sets and the reduced homogeneous ideals which define them. The ''projective varieties'' are the projective algebraic sets whose defining ideal is prime. In other words, a projective variety is a projective algebraic set, whose homogeneous coordinate ring is an integral domain, the ''projective coordinates ring'' being defined as the quotient of the graded ring or the polynomials in variables by the homogeneous (reduced) ideal defining the variety. Every projective algebraic set may be uniquely decomposed into a finite union of projective varieties. The only regular functions which may be defined properly on a projective variety are the constant functions. Thus this notion is not used in projective situations. On the other hand, the ''field of the rational functions'' or ''function field '' is a useful notion, which, similarly to the affine case, is defined as the set of the quotients of two homogeneous elements of the same degree in the homogeneous coordinate ring.


Real algebraic geometry

Real algebraic geometry is the study of the real points of algebraic varieties. The fact that the field of the real numbers is an ordered field cannot be ignored in such a study. For example, the curve of equation x^2+y^2-a=0 is a circle if a>0, but does not have any real point if a<0. It follows that real algebraic geometry is not only the study of the real algebraic varieties, but has been generalized to the study of the ''semi-algebraic sets'', which are the solutions of systems of polynomial equations and polynomial inequalities. For example, a branch of the
hyperbola In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, ca ...
of equation x y-1 = 0 is not an algebraic variety, but is a semi-algebraic set defined by x y-1=0 and x>0 or by x y-1=0 and x+y>0. One of the challenging problems of real algebraic geometry is the unsolved
Hilbert's sixteenth problem Hilbert's 16th problem was posed by David Hilbert at the Paris conference of the International Congress of Mathematicians in 1900, as part of his list of 23 problems in mathematics. The original problem was posed as the ''Problem of the topology ...
: Decide which respective positions are possible for the ovals of a nonsingular plane curve of degree 8.


Computational algebraic geometry

One may date the origin of computational algebraic geometry to meeting EUROSAM'79 (International Symposium on Symbolic and Algebraic Manipulation) held at
Marseille Marseille ( , , ; also spelled in English as Marseilles; oc, Marselha ) is the prefecture of the French department of Bouches-du-Rhône and capital of the Provence-Alpes-Côte d'Azur region. Situated in the camargue region of southern Fra ...
, France, in June 1979. At this meeting, * Dennis S. Arnon showed that George E. Collins's Cylindrical algebraic decomposition (CAD) allows the computation of the topology of semi-algebraic sets, * Bruno Buchberger presented the Gröbner bases and his algorithm to compute them, *
Daniel Lazard Daniel Lazard (born December 10, 1941) is a French mathematician and computer scientist. He is emeritus professor at the University of Paris VI. Career Daniel Lazard was born in Carpentras, in southern France. After his undergraduate educati ...
presented a new algorithm for solving systems of homogeneous polynomial equations with a computational complexity which is essentially polynomial in the expected number of solutions and thus simply exponential in the number of the unknowns. This algorithm is strongly related with Macaulay's multivariate resultant. Since then, most results in this area are related to one or several of these items either by using or improving one of these algorithms, or by finding algorithms whose complexity is simply exponential in the number of the variables. A body of mathematical theory complementary to symbolic methods called numerical algebraic geometry has been developed over the last several decades. The main computational method is
homotopy continuation Numerical algebraic geometry is a field of computational mathematics, particularly computational algebraic geometry, which uses methods from numerical analysis to study and manipulate the solutions of systems of polynomial equations. Homotopy con ...
. This supports, for example, a model of
floating point In computing, floating-point arithmetic (FP) is arithmetic that represents real numbers approximately, using an integer with a fixed precision, called the significand, scaled by an integer exponent of a fixed base. For example, 12.345 can ...
computation for solving problems of algebraic geometry.


Gröbner basis

A Gröbner basis is a system of generators of a polynomial
ideal Ideal may refer to: Philosophy * Ideal (ethics), values that one actively pursues as goals * Platonic ideal, a philosophical idea of trueness of form, associated with Plato Mathematics * Ideal (ring theory), special subsets of a ring considered ...
whose computation allows the deduction of many properties of the affine algebraic variety defined by the ideal. Given an ideal ''I'' defining an algebraic set ''V'': * ''V'' is empty (over an algebraically closed extension of the basis field), if and only if the Gröbner basis for any
monomial order In mathematics, a monomial order (sometimes called a term order or an admissible order) is a total order on the set of all ( monic) monomials in a given polynomial ring, satisfying the property of respecting multiplication, i.e., * If u \leq v and ...
ing is reduced to . * By means of the
Hilbert series In commutative algebra, the Hilbert function, the Hilbert polynomial, and the Hilbert series of a graded commutative algebra finitely generated over a field are three strongly related notions which measure the growth of the dimension of the homog ...
one may compute the
dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coord ...
and the degree of ''V'' from any Gröbner basis of ''I'' for a monomial ordering refining the total degree. * If the dimension of ''V'' is 0, one may compute the points (finite in number) of ''V'' from any Gröbner basis of ''I'' (see Systems of polynomial equations). * A Gröbner basis computation allows one to remove from ''V'' all irreducible components which are contained in a given hypersurface. * A Gröbner basis computation allows one to compute the Zariski closure of the image of ''V'' by the projection on the ''k'' first coordinates, and the subset of the image where the projection is not proper. * More generally Gröbner basis computations allow one to compute the Zariski closure of the image and the critical points of a rational function of ''V'' into another affine variety. Gröbner basis computations do not allow one to compute directly the primary decomposition of ''I'' nor the prime ideals defining the irreducible components of ''V'', but most algorithms for this involve Gröbner basis computation. The algorithms which are not based on Gröbner bases use
regular chain In computer algebra, a regular chain is a particular kind of triangular set in a multivariate polynomial ring over a field. It enhances the notion of characteristic set. Introduction Given a linear system, one can convert it to a triangular ...
s but may need Gröbner bases in some exceptional situations. Gröbner bases are deemed to be difficult to compute. In fact they may contain, in the worst case, polynomials whose degree is doubly exponential in the number of variables and a number of polynomials which is also doubly exponential. However, this is only a worst case complexity, and the complexity bound of Lazard's algorithm of 1979 may frequently apply. Faugère F5 algorithm realizes this complexity, as it may be viewed as an improvement of Lazard's 1979 algorithm. It follows that the best implementations allow one to compute almost routinely with algebraic sets of degree more than 100. This means that, presently, the difficulty of computing a Gröbner basis is strongly related to the intrinsic difficulty of the problem.


Cylindrical algebraic decomposition (CAD)

CAD is an algorithm which was introduced in 1973 by G. Collins to implement with an acceptable complexity 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 defi ...
on quantifier elimination over the real numbers. This theorem concerns the formulas of the first-order logic whose atomic formulas are polynomial equalities or inequalities between polynomials with real coefficients. These formulas are thus the formulas which may be constructed from the atomic formulas by the logical operators ''and'' (∧), ''or'' (∨), ''not'' (¬), ''for all'' (∀) and ''exists'' (∃). Tarski's theorem asserts that, from such a formula, one may compute an equivalent formula without quantifier (∀, ∃). The complexity of CAD is doubly exponential in the number of variables. This means that CAD allows, in theory, to solve every problem of real algebraic geometry which may be expressed by such a formula, that is almost every problem concerning explicitly given varieties and semi-algebraic sets. While Gröbner basis computation has doubly exponential complexity only in rare cases, CAD has almost always this high complexity. This implies that, unless if most polynomials appearing in the input are linear, it may not solve problems with more than four variables. Since 1973, most of the research on this subject is devoted either to improving CAD or finding alternative algorithms in special cases of general interest. As an example of the state of art, there are efficient algorithms to find at least a point in every connected component of a semi-algebraic set, and thus to test if a semi-algebraic set is empty. On the other hand, CAD is yet, in practice, the best algorithm to count the number of connected components.


Asymptotic complexity vs. practical efficiency

The basic general algorithms of computational geometry have a double exponential worst case complexity. More precisely, if ''d'' is the maximal degree of the input polynomials and ''n'' the number of variables, their complexity is at most d^ for some constant ''c'', and, for some inputs, the complexity is at least d^ for another constant ''c''′. During the last 20 years of the 20th century, various algorithms have been introduced to solve specific subproblems with a better complexity. Most of these algorithms have a complexity d^. Among these algorithms which solve a sub problem of the problems solved by Gröbner bases, one may cite ''testing if an affine variety is empty'' and ''solving nonhomogeneous polynomial systems which have a finite number of solutions.'' Such algorithms are rarely implemented because, on most entries Faugère's F4 and F5 algorithms have a better practical efficiency and probably a similar or better complexity (''probably'' because the evaluation of the complexity of Gröbner basis algorithms on a particular class of entries is a difficult task which has been done only in a few special cases). The main algorithms of real algebraic geometry which solve a problem solved by CAD are related to the topology of semi-algebraic sets. One may cite ''counting the number of connected components'', ''testing if two points are in the same components'' or ''computing a Whitney stratification of a real algebraic set''. They have a complexity of d^, but the constant involved by ''O'' notation is so high that using them to solve any nontrivial problem effectively solved by CAD, is impossible even if one could use all the existing computing power in the world. Therefore, these algorithms have never been implemented and this is an active research area to search for algorithms with have together a good asymptotic complexity and a good practical efficiency.


Abstract modern viewpoint

The modern approaches to algebraic geometry redefine and effectively extend the range of basic objects in various levels of generality to schemes, formal schemes,
ind-scheme In algebraic geometry, an ind-scheme is a set-valued functor that can be written (represented) as a direct limit (i.e., inductive limit) of closed embedding of schemes. Examples *\mathbbP^ = \varinjlim \mathbbP^N is an ind-scheme. *Perhaps th ...
s, algebraic spaces, algebraic stacks and so on. The need for this arises already from the useful ideas within theory of varieties, e.g. the formal functions of Zariski can be accommodated by introducing nilpotent elements in structure rings; considering spaces of loops and arcs, constructing quotients by group actions and developing formal grounds for natural intersection theory and deformation theory lead to some of the further extensions. Most remarkably, in the late 1950s, algebraic varieties were subsumed into Alexander Grothendieck's concept of a
scheme A scheme is a systematic plan for the implementation of a certain idea. Scheme or schemer may refer to: Arts and entertainment * ''The Scheme'' (TV series), a BBC Scotland documentary series * The Scheme (band), an English pop band * ''The Schem ...
. Their local objects are affine schemes or prime spectra which are locally ringed spaces which form a category which is antiequivalent to the category of commutative unital rings, extending the duality between the category of affine algebraic varieties over a field ''k'', and the category of finitely generated reduced ''k''-algebras. The gluing is along Zariski topology; one can glue within the category of locally ringed spaces, but also, using the Yoneda embedding, within the more abstract category of presheaves of sets over the category of affine schemes. The Zariski topology in the set theoretic sense is then replaced by a Grothendieck topology. Grothendieck introduced Grothendieck topologies having in mind more exotic but geometrically finer and more sensitive examples than the crude Zariski topology, namely the
étale topology In algebraic geometry, the étale topology is a Grothendieck topology on the category of schemes which has properties similar to the Euclidean topology, but unlike the Euclidean topology, it is also defined in positive characteristic. The étale to ...
, and the two flat Grothendieck topologies: fppf and fpqc; nowadays some other examples became prominent including
Nisnevich topology In algebraic geometry, the Nisnevich topology, sometimes called the completely decomposed topology, is a Grothendieck topology on the category of schemes which has been used in algebraic K-theory, A¹ homotopy theory, and the theory of motives. I ...
. Sheaves can be furthermore generalized to stacks in the sense of Grothendieck, usually with some additional representability conditions leading to Artin stacks and, even finer, Deligne–Mumford stacks, both often called algebraic stacks. Sometimes other algebraic sites replace the category of affine schemes. For example,
Nikolai Durov Nikolai Valeryevich Durov (russian: Никола́й Вале́рьевич Ду́ров; born 21 November 1980) is a Russian programmer and mathematician. He is the elder brother of Pavel Durov, with whom he founded the social networking site ...
has introduced commutative algebraic monads as a generalization of local objects in a generalized algebraic geometry. Versions of a tropical geometry, of an absolute geometry over a field of one element and an algebraic analogue of Arakelov's geometry were realized in this setup. Another formal generalization is possible to universal algebraic geometry in which every variety of algebras has its own algebraic geometry. The term ''variety of algebras'' should not be confused with ''algebraic variety''. The language of schemes, stacks and generalizations has proved to be a valuable way of dealing with geometric concepts and became cornerstones of modern algebraic geometry. Algebraic stacks can be further generalized and for many practical questions like deformation theory and intersection theory, this is often the most natural approach. One can extend the
Grothendieck site In category theory, a branch of mathematics, a Grothendieck topology is a structure on a category ''C'' that makes the objects of ''C'' act like the open sets of a topological space. A category together with a choice of Grothendieck topology is ca ...
of affine schemes to a higher categorical site of derived affine schemes, by replacing the commutative rings with an infinity category of differential graded commutative algebras, or of simplicial commutative rings or a similar category with an appropriate variant of a Grothendieck topology. One can also replace presheaves of sets by presheaves of simplicial sets (or of infinity groupoids). Then, in presence of an appropriate homotopic machinery one can develop a notion of derived stack as such a presheaf on the infinity category of derived affine schemes, which is satisfying certain infinite categorical version of a sheaf axiom (and to be algebraic, inductively a sequence of representability conditions). Quillen model categories, Segal categories and quasicategories are some of the most often used tools to formalize this yielding the ''
derived algebraic geometry Derived algebraic geometry is a branch of mathematics that generalizes algebraic geometry to a situation where commutative rings, which provide local charts, are replaced by either differential graded algebras (over \mathbb), simplicial commutativ ...
'', introduced by the school of Carlos Simpson, including Andre Hirschowitz, Bertrand Toën, Gabrielle Vezzosi, Michel Vaquié and others; and developed further by Jacob Lurie, Bertrand Toën, and Gabriele Vezzosi. Another (noncommutative) version of derived algebraic geometry, using A-infinity categories has been developed from the early 1990s by Maxim Kontsevich and followers.


History


Before the 16th century

Some of the roots of algebraic geometry date back to the work of the Hellenistic Greeks from the 5th century BC. The Delian problem, for instance, was to construct a length ''x'' so that the cube of side ''x'' contained the same volume as the rectangular box ''a''2''b'' for given sides ''a'' and ''b''.
Menaechmus :''There is also a Menaechmus in Plautus' play, ''The Menaechmi''.'' Menaechmus ( el, Μέναιχμος, 380–320 BC) was an ancient Greek mathematician, geometer and philosopher born in Alopeconnesus or Prokonnesos in the Thracian Chersones ...
() considered the problem geometrically by intersecting the pair of plane conics ''ay'' = ''x''2 and ''xy'' = ''ab''. In the 3rd century BC, Archimedes and Apollonius systematically studied additional problems on conic sections using coordinates. Apollonius in the Conics further developed a method that is so similar to analytic geometry that his work is sometimes thought to have anticipated the work of Descartes by some 1800 years.His application of reference lines, a
diameter In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle. Both definitions are also valid f ...
and a
tangent In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. Mo ...
is essentially no different from our modern use of a coordinate frame, where the distances measured along the diameter from the point of tangency are the abscissas, and the segments parallel to the tangent and intercepted between the axis and the curve are the ordinates. He further developed relations between the abscissas and the corresponding cordinates using geometric methods like using parabolas and curves. Medieval mathematicians, including Omar Khayyam, leonardo of Pisa, Gersonides and Nicole Oresme in the
Medieval Period In the history of Europe, the Middle Ages or medieval period lasted approximately from the late 5th to the late 15th centuries, similar to the post-classical period of global history. It began with the fall of the Western Roman Empire a ...
, solved certain cubic and quadratic equations by purely algebraic means and then interpreted the results geometrically. The Persian mathematician
Omar Khayyám Ghiyāth al-Dīn Abū al-Fatḥ ʿUmar ibn Ibrāhīm Nīsābūrī (18 May 1048 – 4 December 1131), commonly known as Omar Khayyam ( fa, عمر خیّام), was a polymath, known for his contributions to mathematics, astronomy, philosophy, an ...
(born 1048 AD) believed that there was a relationship between
arithmetic Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers— addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th ...
,
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary ...
and
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
. This was criticized by Jeffrey Oaks, who claims that the study of curves by means of equations originated with Descartes in the seventeenth century.


Renaissance

Such techniques of applying geometrical constructions to algebraic problems were also adopted by a number of
Renaissance The Renaissance ( , ) , from , with the same meanings. is a period in European history marking the transition from the Middle Ages to modernity and covering the 15th and 16th centuries, characterized by an effort to revive and surpass ide ...
mathematicians such as
Gerolamo Cardano Gerolamo Cardano (; also Girolamo or Geronimo; french: link=no, Jérôme Cardan; la, Hieronymus Cardanus; 24 September 1501– 21 September 1576) was an Italian polymath, whose interests and proficiencies ranged through those of mathematician, ...
and Niccolò Fontana "Tartaglia" on their studies of the cubic equation. The geometrical approach to construction problems, rather than the algebraic one, was favored by most 16th and 17th century mathematicians, notably Blaise Pascal who argued against the use of algebraic and analytical methods in geometry. The French mathematicians Franciscus Vieta and later
René Descartes René Descartes ( or ; ; Latinized: Renatus Cartesius; 31 March 1596 – 11 February 1650) was a French philosopher, scientist, and mathematician, widely considered a seminal figure in the emergence of modern philosophy and science. Ma ...
and
Pierre de Fermat Pierre de Fermat (; between 31 October and 6 December 1607 – 12 January 1665) was a French mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality. In particular, he ...
revolutionized the conventional way of thinking about construction problems through the introduction of coordinate geometry. They were interested primarily in the properties of ''algebraic curves'', such as those defined by Diophantine equations (in the case of Fermat), and the algebraic reformulation of the classical Greek works on conics and cubics (in the case of Descartes). During the same period, Blaise Pascal and
Gérard Desargues Girard Desargues (; 21 February 1591 – September 1661) was a French mathematician and engineer, who is considered one of the founders of projective geometry. Desargues' theorem, the Desargues graph, and the crater Desargues on the Moon a ...
approached geometry from a different perspective, developing the
synthetic Synthetic things are composed of multiple parts, often with the implication that they are artificial. In particular, 'synthetic' may refer to: Science * Synthetic chemical or compound, produced by the process of chemical synthesis * Synthetic ...
notions of projective geometry. Pascal and Desargues also studied curves, but from the purely geometrical point of view: the analog of the Greek ''ruler and compass construction''. Ultimately, the
analytic geometry In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry. Analytic geometry is used in physics and enginee ...
of Descartes and Fermat won out, for it supplied the 18th century mathematicians with concrete quantitative tools needed to study physical problems using the new calculus of Newton and
Leibniz Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of ma ...
. However, by the end of the 18th century, most of the algebraic character of coordinate geometry was subsumed by the ''calculus of infinitesimals'' of Lagrange and Euler.


19th and early 20th century

It took the simultaneous 19th century developments of non-Euclidean geometry and Abelian integrals in order to bring the old algebraic ideas back into the geometrical fold. The first of these new developments was seized up by Edmond Laguerre and Arthur Cayley, who attempted to ascertain the generalized metric properties of projective space. Cayley introduced the idea of ''homogeneous polynomial forms'', and more specifically quadratic forms, on projective space. Subsequently, Felix Klein studied projective geometry (along with other types of geometry) from the viewpoint that the geometry on a space is encoded in a certain class of transformations on the space. By the end of the 19th century, projective geometers were studying more general kinds of transformations on figures in projective space. Rather than the projective linear transformations which were normally regarded as giving the fundamental Kleinian geometry on projective space, they concerned themselves also with the higher degree birational transformations. This weaker notion of congruence would later lead members of the 20th century
Italian school of algebraic geometry In relation to the history of mathematics, the Italian school of algebraic geometry refers to mathematicians and their work in birational geometry, particularly on algebraic surfaces, centered around Rome roughly from 1885 to 1935. There were 30 ...
to classify
algebraic surface In mathematics, an algebraic surface is an algebraic variety of dimension two. In the case of geometry over the field of complex numbers, an algebraic surface has complex dimension two (as a complex manifold, when it is non-singular) and so of di ...
s up to birational isomorphism. The second early 19th century development, that of Abelian integrals, would lead
Bernhard Riemann Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first ...
to the development of
Riemann surface In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed ver ...
s. In the same period began the algebraization of the algebraic geometry through
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. Prom ...
. The prominent results in this direction are Hilbert's basis theorem and Hilbert's Nullstellensatz, which are the basis of the connexion between algebraic geometry and commutative algebra, and Macaulay's multivariate resultant, which is the basis of elimination theory. Probably because of the size of the computation which is implied by multivariate resultants, elimination theory was forgotten during the middle of the 20th century until it was renewed by singularity theory and computational algebraic geometry.


20th century

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 ...
, Oscar Zariski and André Weil developed a foundation for algebraic geometry based on contemporary
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. Prom ...
, including
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 the theory of ideals. One of the goals was to give a rigorous framework for proving the results of
Italian school of algebraic geometry In relation to the history of mathematics, the Italian school of algebraic geometry refers to mathematicians and their work in birational geometry, particularly on algebraic surfaces, centered around Rome roughly from 1885 to 1935. There were 30 ...
. In particular, this school used systematically the notion of generic point without any precise definition, which was first given by these authors during the 1930s. In the 1950s and 1960s, Jean-Pierre Serre and Alexander Grothendieck recast the foundations making use of sheaf theory. Later, from about 1960, and largely led by Grothendieck, the idea of schemes was worked out, in conjunction with a very refined apparatus of homological techniques. After a decade of rapid development the field stabilized in the 1970s, and new applications were made, both to
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Ma ...
and to more classical geometric questions on algebraic varieties, singularities, moduli, and formal moduli. An important class of varieties, not easily understood directly from their defining equations, are the abelian varieties, which are the projective varieties whose points form an abelian group. The prototypical examples are the
elliptic curve In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. I ...
s, which have a rich theory. They were instrumental in the proof of Fermat's Last Theorem and are also used in elliptic-curve cryptography. In parallel with the abstract trend of the algebraic geometry, which is concerned with general statements about varieties, methods for effective computation with concretely-given varieties have also been developed, which lead to the new area of computational algebraic geometry. One of the founding methods of this area is the theory of Gröbner bases, introduced by Bruno Buchberger in 1965. Another founding method, more specially devoted to real algebraic geometry, is the cylindrical algebraic decomposition, introduced by George E. Collins in 1973. See also:
derived algebraic geometry Derived algebraic geometry is a branch of mathematics that generalizes algebraic geometry to a situation where commutative rings, which provide local charts, are replaced by either differential graded algebras (over \mathbb), simplicial commutativ ...
.


Analytic geometry

An
analytic variety In mathematics, and in particular differential geometry and complex geometry, a complex analytic variety Complex analytic variety (or just variety) is sometimes required to be irreducible and (or) reduced or complex analytic space is a generali ...
is defined locally as the set of common solutions of several equations involving
analytic function In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex ...
s. It is analogous to the included concept of real or complex
algebraic variety Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. ...
. Any complex manifold is an analytic variety. Since analytic varieties may have singular points, not all analytic varieties are manifolds. Modern analytic geometry is essentially equivalent to real and complex algebraic geometry, as has been shown by Jean-Pierre Serre in his paper '' GAGA'', the name of which is French for ''Algebraic geometry and analytic geometry''. Nevertheless, the two fields remain distinct, as the methods of proof are quite different and algebraic geometry includes also geometry in finite characteristic.


Applications

Algebraic geometry now finds applications in
statistics Statistics (from German: '' Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, indust ...
,
control theory Control theory is a field of mathematics that deals with the control system, control of dynamical systems in engineered processes and machines. The objective is to develop a model or algorithm governing the application of system inputs to drive ...
,
robotics Robotics is an interdisciplinary branch of computer science and engineering. Robotics involves design, construction, operation, and use of robots. The goal of robotics is to design machines that can help and assist humans. Robotics integrat ...
,
error-correcting codes In computing, telecommunication, information theory, and coding theory, an error correction code, sometimes error correcting code, (ECC) is used for controlling errors in data over unreliable or noisy communication channels. The central idea ...
,
phylogenetics In biology, phylogenetics (; from Greek φυλή/ φῦλον [] "tribe, clan, race", and wikt:γενετικός, γενετικός [] "origin, source, birth") is the study of the evolutionary history and relationships among or within groups ...
and geometric modelling. There are also connections to string theory,
game theory Game theory is the study of mathematical models of strategic interactions among rational agents. Myerson, Roger B. (1991). ''Game Theory: Analysis of Conflict,'' Harvard University Press, p.&nbs1 Chapter-preview links, ppvii–xi It has appli ...
, graph matchings,
soliton In mathematics and physics, a soliton or solitary wave is a self-reinforcing wave packet that maintains its shape while it propagates at a constant velocity. Solitons are caused by a cancellation of nonlinear and dispersive effects in the me ...
s and integer programming.


See also

* Glossary of classical algebraic geometry * Important publications in algebraic geometry * List of algebraic surfaces * Noncommutative algebraic geometry


Notes


References


Sources

*


Further reading

;Some classic textbooks that predate schemes: * * * * ;Modern textbooks that do not use the language of schemes: * * * * * * ;Textbooks in computational algebraic geometry * * * * * * * * ;Textbooks and references for schemes: * * * * * *


External links


''Foundations of Algebraic Geometry'' by Ravi Vakil, 808 pp.


entry o
PlanetMath

English translation of the van der Waerden textbook
*
The Stacks Project
an open source textbook and reference work on algebraic stacks and algebraic geometry {{DEFAULTSORT:Algebraic Geometry Fields of mathematics