A
plane curve defined by an
implicit equation :F(x,y)=0, where is a
smooth function is said to be
singular at a point if the
Taylor series of has
order at least at this point. The reason for this is that, in
differential calculus, 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 :F(x,y,z,\ldots) = 0 the
singular points are those at which all the
partial derivatives simultaneously vanish. A general
algebraic variety being defined as the common zeros of several
polynomials, the condition on a point of to be a singular point is that the
Jacobian matrix of the first-order partial derivatives of the polynomials has a
rank 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 and
dense in the variety (for the
Zariski topology, as well as for the usual topology, in the case of varieties defined over the
complex numbers). 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 near every regular point. But a real variety may be a manifold and have singular points. For example the equation defines a real
analytic manifold but has a singular point at the origin. This may be explained by saying that the curve has two
complex conjugate branches that cut the real branch at the origin. ==Singular points of smooth mappings==