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A curve with a triple point at the origin: x(t) = sin(2t) + cos(t), y(t) = sin(t) + cos(2t) In general, if all the terms of degree less than k are 0, and at least one term of degree k is not 0 in f, then curve is said to have a multiple point of order k or a k-ple point.
This presents no difficulty for a an ordinary point (Lazarus Fuchs 1866). When a is a regular singular point, which by definition means that has a pole of order at most i at a, the Frobenius method also can be made to work and provide n independent solutions near a. Otherwise the point a is an irregular singularity.
Points of V 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). [1]
In mathematics, an ordinary singularity of an algebraic curve is a singular point of multiplicity r where the r tangents at the point are distinct (Walker 1950, p. 54). In higher dimensions the literature on algebraic geometry contains many inequivalent definitions of ordinary singular points.
For affine and projective varieties, the singularities are the points where the Jacobian matrix has a rank which is lower than at other points of the variety. An equivalent definition in terms of commutative algebra may be given, which extends to abstract varieties and schemes : A point is singular if the local ring at this point is not a ...
Otherwise it is called a singular point. The tangent space has an interpretation in terms of K[t]/(t 2), the dual numbers for K; in the parlance of schemes, morphisms from Spec K[t]/(t 2) to a scheme X over K correspond to a choice of a rational point x ∈ X(k) and an element of the tangent space at x. [3] Therefore, one also talks about ...
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Singularity (system theory), in dynamical and social systems, a context in which a small change can cause a large effect Gravitational singularity, in general relativity, a point in which gravity is so intense that spacetime itself becomes ill-defined