In the

mathematical Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

field of

algebraic geometry Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...

, a singular point of an

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 solution set, set of solutions of a system of polynomial equations over the real number, ...

is a point that is 'special' (so, singular), in the geometric sense that at this point the

tangent space In mathematics, the tangent space of a manifold is a generalization of to curves in two-dimensional space and to surfaces in three-dimensional space in higher dimensions. In the context of physics the tangent space to a manifold at a point can be ...

at the variety may not be regularly defined. In case of varieties defined over the reals, this notion generalizes the notion of local non-flatness. A point of an algebraic variety that is not singular is said to be regular. An algebraic variety that has no singular point is said to be non-singular or smooth. The concept is generalized to

smooth scheme In algebraic geometry, a smooth scheme over a field is a scheme which is well approximated by affine space near any point. Smoothness is one way of making precise the notion of a scheme with no singular points. A special case is the notion of a sm ...

s in the modern language of

scheme theory In mathematics, specifically algebraic geometry, a scheme is a structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities (the equations and define the same algebraic variety but different s ...

Definition

plane curve In mathematics, a plane curve is a curve in a plane that may be a Euclidean plane, an affine plane or a projective plane. The most frequently studied cases are smooth plane curves (including piecewise smooth plane curves), and algebraic plane c ...

defined by an

implicit equation In mathematics, an implicit equation is a relation of the form R(x_1, \dots, x_n) = 0, where is a function of several variables (often a polynomial). For example, the implicit equation of the unit circle is x^2 + y^2 - 1 = 0. An implicit func ...

F(x,y)=0,

where is a

smooth function In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives (''differentiability class)'' it has over its domain. A function of class C^k is a function of smoothness at least ; t ...

is said to be ''singular'' at a point if the

Taylor series In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...

of has

order Order, ORDER or Orders may refer to: * A socio-political or established or existing order, e.g. World order, Ancien Regime, Pax Britannica * Categorization, the process in which ideas and objects are recognized, differentiated, and understood ...

at least at this point. The reason for this is that, in

differential calculus In mathematics, differential calculus is a subfield of calculus that studies the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus—the study of the area beneath a curve. ...

, the tangent at the point of such a curve is defined by the equation :

(x-x_0)F'_x(x_0,y_0) + (y-y_0)F'_y(x_0,y_0)=0,

whose left-hand side is the term of degree one of the Taylor expansion. Thus, if this term is zero, the tangent may not be defined in the standard way, either because it does not exist or a special definition must be provided. In general for a

hypersurface In geometry, a hypersurface is a generalization of the concepts of hyperplane, plane curve, and surface. A hypersurface is a manifold or an algebraic variety of dimension , which is embedded in an ambient space of dimension , generally a Euclidea ...

F(x,y,z,\ldots) = 0

the singular points are those at which all the

partial derivative In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). P ...

s simultaneously vanish. A general

being defined as the common zeros of several

polynomial In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...

s, the condition on a point of to be a singular point is that the

Jacobian matrix In vector calculus, the Jacobian matrix (, ) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. If this matrix is square, that is, if the number of variables equals the number of component ...

of the first-order partial derivatives of the polynomials has a

rank A rank is a position in a hierarchy. It can be formally recognized—for example, cardinal, chief executive officer, general, professor—or unofficial. People Formal ranks * Academic rank * Corporate title * Diplomatic rank * Hierarchy ...

at that is lower than the rank at other points of the variety. Points of that are not singular are called non-singular or regular. It is always true that almost all points are non-singular, in the sense that the non-singular points form a set that is both

open Open or OPEN may refer to: Music * Open (band), Australian pop/rock band * The Open (band), English indie rock band * ''Open'' (Blues Image album), 1969 * ''Open'' (Gerd Dudek, Buschi Niebergall, and Edward Vesala album), 1979 * ''Open'' (Go ...

and

dense Density (volumetric mass density or specific mass) is the ratio of a substance's mass to its volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' (or ''d'') can also be use ...

in the variety (for the

Zariski topology In algebraic geometry and commutative algebra, the Zariski topology is a topology defined on geometric objects called varieties. It is very different from topologies that are commonly used in real or complex analysis; in particular, it is not ...

, as well as for the usual topology, in the case of varieties defined over the

complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...

s). In case of a real variety (that is the set of the points with real coordinates of a variety defined by polynomials with real coefficients), the variety is a

manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a N ...

near every regular point. But it is important to note that a real variety may be a manifold and have singular points. For example the equation defines a real

analytic manifold In mathematics, an analytic manifold, also known as a C^\omega manifold, is a differentiable manifold with analytic transition maps. The term usually refers to real analytic manifolds, although complex manifolds are also analytic. In algebraic geo ...

but has a singular point at the origin. This may be explained by saying that the curve has two

complex conjugate In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, if a and b are real numbers, then the complex conjugate of a + bi is a - ...

branch A branch, also called a ramus in botany, is a stem that grows off from another stem, or when structures like veins in leaves are divided into smaller veins. History and etymology In Old English, there are numerous words for branch, includ ...

es that cut the real branch at the origin.

Singular points of smooth mappings

As the notion of singular points is a purely local property, the above definition can be extended to cover the wider class of smooth mappings (functions from to where all derivatives exist). Analysis of these singular points can be reduced to the algebraic variety case by considering the jets of the mapping. The th jet is the

of the mapping truncated at degree and deleting the

constant term In mathematics, a constant term (sometimes referred to as a free term) is a term in an algebraic expression that does not contain any variables and therefore is constant. For example, in the quadratic polynomial, :x^2 + 2x + 3,\ The number 3 i ...

Nodes

In classical algebraic geometry, certain special singular points were also called nodes. A node is a singular point where the

Hessian matrix In mathematics, the Hessian matrix, Hessian or (less commonly) Hesse matrix is a square matrix of second-order partial derivatives of a scalar-valued Function (mathematics), function, or scalar field. It describes the local curvature of a functio ...

is non-singular; this implies that the singular point has multiplicity two and the tangent cone is not singular outside its vertex.

References

{{reflist Algebraic varieties Singularity theory

Definition

Singular points of smooth mappings

Nodes

See also

References