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An algebraic curve in the Euclidean plane is the set of the points whose coordinates are the solutions of a bivariate polynomial equation p(x, y) = 0.This equation is often called the implicit equation of the curve, in contrast to the curves that are the graph of a function defining explicitly y as a function of x.
Algebraic curves in the plane may be defined as the set of points (x, y) satisfying an equation of the form (,) =, where f is a polynomial function :. If f is expanded as = + + + + + + If the origin (0, 0) is on the curve then a 0 = 0.
A space curve is a curve for which is at least three-dimensional; a skew curve is a space curve which lies in no plane. These definitions of plane, space and skew curves apply also to real algebraic curves , although the above definition of a curve does not apply (a real algebraic curve may be disconnected ).
Algebraic varieties of dimension one are called algebraic curves and algebraic varieties of dimension two are called algebraic surfaces. In the context of modern scheme theory, an algebraic variety over a field is an integral (irreducible and reduced) scheme over that field whose structure morphism is separated and of finite type.
Algebraic curves. Rational curves. Rational curves are subdivided according to the degree of the polynomial. Degree 1. Line; Degree 2. Plane curves ...
When is an algebraic curve with field of definition the complex numbers, and if has no singular points, then these definitions agree and coincide with the topological definition applied to the Riemann surface of (its manifold of complex points).
The simplest example of singularities are curves that cross themselves. But there are other types of singularities, like cusps. For example, the equation y 2 − x 3 = 0 defines a curve that has a cusp at the origin x = y = 0. One could define the x-axis as a tangent at this point, but this definition can not be the same as the definition at ...
A Hurwitz curve is a complex algebraic curve of genus g>0 with the maximum possible number 84(g–1) of automorphisms. hyperbolism Essentially a blow-up of a curve at a point. See Salmon (1879, p.175). hypercusp A singularity of a curve of some multiplicity r whose tangent cone is a single line meeting the curve with order r+1. (Coolidge 1931 ...
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