What is "algebraic curve"

Algebraic \Al`ge*bra"ic\, Algebraical \Al`ge*bra"ic*al\, a.

1. Of or pertaining to algebra; using algebra; according to the laws of algebra; containing an operation of algebra, or deduced from such operation; as, algebraic characters; algebraical writings; algebraic geometry.

2. progressing by constant multiplicatory factors; -- of a series of numbers. Contrasted to arithmetical. algebraic progression

   Syn: algebraic

   Algebraic curve, a curve such that the equation which expresses the relation between the co["o]rdinates of its points involves only the ordinary operations of algebra; -- opposed to a transcendental curve.

In mathematics, an algebraic curve or plane algebraic curve is the set of points on 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 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 is the set of points of whose coordinates are zeros of some bivariate polynomial with coefficients in , where is some algebraically closed extension of . The points of the curve with coordinates in are the -points of the curve and, all together, are the 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 as to only the real points, the exact meaning usually clear from 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.