Algebraic Curve

Algebraic Curve

In mathematics, algebraic curves are the simplest objects of Euclidean geometry that can not be defined by linear properties. Specifically, in Euclidean geometry, a plane algebraic curve is the set of the points of the Euclidean plane whose coordinates are zeros of some polynomial in two variables.

For example, the unit circle is an algebraic curve, being the set of zeros of the polynomial

Various technical considerations have led to consider that the complex zeros of a polynomial belong to the curve. Also, the notion of algebraic curve has been generalized to allow the coefficients of the defining polynomial and the coordinates of the points of the curve to belong to any field, leading to the following definition.

In algebraic geometry, a plane affine algebraic curve defined over a field k is the set of points of K2 whose coordinates are zeros of some bivariate polynomial with coefficients in k, where K is some algebraically closed extension of k. The points of the curve with coordinates in k are the k-points of the curve and, all together, are the k part of the curve.

For example, is a point of the curve defined by and the usual unit circle is the real part of this curve. The term "unit circle" may refer to all the complex points as well to only the real points, the exact meaning being usually clear form the context. The equation defines an algebraic curve, whose real part is empty.

More generally, one may consider algebraic curves that are not contained in the plane, but in a space of higher dimension. A curve that is not contained in some plane is called a skew curve. The simplest example of a skew algebraic curve is the twisted cubic. One may also consider algebraic curves contained in the projective space and even algebraic curves that are defined independently to any embedding in an affine or projective space. This leads to the most general definition of an algebraic curve:

In algebraic geometry, an algebraic curve is an algebraic variety of dimension one.