Algebraic curves
In mathematics, a plane real algebraic curve is the set of points on the Euclidean plane whose coordinates are zeros of some polynomial in two variables. More generally an algebraic curve is similar but may be embedded in a higher dimensional space or defined over some more general field.
For example, the unit circle is a real algebraic curve, being the set of zeros of the polynomial x2 + y2 – 1.
Various technical considerations result in the complex zeros of a polynomial being considered as belonging 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, (2,√−3) is a point of the curve defined by x2 + y2 − 1 = 0 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 as to only the real points, the exact meaning usually clear from the context. The equation x2 + y2 + 1 = 0 defines an algebraic curve, whose real part is empty.
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