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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.
Repeatedly blowing up the singular points of a curve will eventually resolve the singularities. The main task with this method is to find a way to measure the complexity of a singularity and to show that blowing up improves this measure. There are many ways to do this. For example, one can use the arithmetic genus of the curve.
A plane curve defined by an implicit equation (,) =,where F is a smooth function is said to be singular at a point if the Taylor series of F has order at least 2 at this point.. The reason for this is that, in differential calculus, the tangent at the point (x 0, y 0) of such a curve is defined by the equation
One could define the x-axis as a tangent at this point, but this definition can not be the same as the definition at other points. In fact, in this case, the x-axis is a "double tangent." 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.
Point a is an ordinary point when functions p 1 (x) and p 0 (x) are analytic at x = a. Point a is a regular singular point if p 1 (x) has a pole up to order 1 at x = a and p 0 has a pole of order up to 2 at x = a. Otherwise point a is an irregular singular point.
Consider a smooth real-valued function of two variables, say f (x, y) where x and y are real numbers.So f is a function from the plane to the line. The space of all such smooth functions is acted upon by the group of diffeomorphisms of the plane and the diffeomorphisms of the line, i.e. diffeomorphic changes of coordinate in both the source and the target.
The study of the analytic structure of an algebraic curve in the neighborhood of a singular point provides accurate information of the topology of singularities. In fact, near a singular point, a real algebraic curve is the union of a finite number of branches that intersect only at the singular point and look either as a cusp or as a smooth curve.
Singularity functions have been heavily studied in the field of mathematics under the alternative names of generalized functions and distribution theory. [ 1 ] [ 2 ] [ 3 ] The functions are notated with brackets, as x − a n {\displaystyle \langle x-a\rangle ^{n}} where n is an integer.