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Algebraic curve

Algebraic curves are the curves considered in algebraic geometry. A plane algebraic curve is the set of the points of coordinates x, y such that f (x, y) = 0, where f is a polynomial in two variables defined over some field F. One says that the curve is defined over F. Algebraic geometry normally considers not only points with coordinates in F but all the points with coordinates in an algebraically closed field K.

If C is a curve defined by a polynomial f with coefficients in F, the curve is said to be defined over F.

In the case of a curve defined over the real numbers, one normally considers points with complex coordinates. In this case, a point with real coordinates is a real point, and the set of all real points is the real part of the curve. It is therefore only the real part of an algebraic curve that can be a topological curve (this is not always the case, as the real part of an algebraic curve may be disconnected and contain isolated points). The whole curve, that is the set of its complex point is, from the topological point of view a surface. In particular, the nonsingular complex projective algebraic curves are called Riemann surfaces.

The points of a curve C with coordinates in a field G are said to be rational over G and can be denoted C (G). When G is the field of the rational numbers, one simply talks of rational points. For example, Fermat's Last Theorem may be restated as: For n > 2, every rational point of the Fermat curve of degree n has a zero coordinate.

Algebraic curves can also be space curves, or curves in a space of higher dimension, say n. They are defined as algebraic varieties of dimension one. They may be obtained as the common solutions of at least n –1 polynomial equations in n variables. If n –1 polynomials are sufficient to define a curve in a space of dimension n, the curve is said to be a complete intersection. By eliminating variables (by any tool of elimination theory), an algebraic curve may be projected onto a plane algebraic curve, which however may introduce new singularities such as cusps or double points.

A plane curve may also be completed to a curve in the projective plane: if a curve is defined by a polynomial f of total degree d, then w f (u / w, v / w) simplifies to a homogeneous polynomial g (u, v, w) of degree d. The values of u, v, w such that g (u, v, w) = 0 are the homogeneous coordinates of the points of the completion of the curve in the projective plane and the points of the initial curve are those such that w is not zero. An example is the Fermat curve u + v = w, which has an affine form x + y = 1. A similar process of homogenization may be defined for curves in higher dimensional spaces.

Except for lines, the simplest examples of algebraic curves are the conics, which are nonsingular curves of degree two and genus zero. Elliptic curves, which are nonsingular curves of genus one, are studied in number theory, and have important applications to cryptography.