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This is a list of Wikipedia articles about curves used in different fields: ... Rational curves are subdivided according to the degree of the polynomial. Degree 1
In mathematics, the rational normal curve is a smooth, rational curve C of degree n in projective n-space P n. It is a simple example of a projective variety; formally, it is the Veronese variety when the domain is the projective line. For n = 2 it is the plane conic Z 0 Z 2 = Z 2 1, and for n = 3 it is the twisted cubic.
Rational Bézier curve – polynomial curve defined in homogeneous coordinates (blue) and its projection on plane – rational curve (red) In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work Der barycentrische Calcul, [1] [2] [3] are a system of coordinates used in projective geometry, just as Cartesian coordinates are used ...
A variety is uniruled if it is covered by a family of rational curves. (More precisely, a variety X {\displaystyle X} is uniruled if there is a variety Y {\displaystyle Y} and a dominant rational map Y × P 1 → X {\displaystyle Y\times \mathbf {P} ^{1}\to X} which does not factor through the projection to Y {\displaystyle Y} .)
Faltings's theorem is a result in arithmetic geometry, according to which a curve of genus greater than 1 over the field of rational numbers has only finitely many rational points. This was conjectured in 1922 by Louis Mordell , [ 1 ] and known as the Mordell conjecture until its 1983 proof by Gerd Faltings . [ 2 ]
In mathematics, the rank of an elliptic curve is the rational Mordell–Weil rank of an elliptic curve defined over the field of rational numbers or more generally a number field K. Mordell's theorem (generalized to arbitrary number fields by André Weil ) says the group of rational points on an elliptic curve has a finite basis .
Gröbner bases are primarily defined for ideals in a polynomial ring = [, …,] over a field K.Although the theory works for any field, most Gröbner basis computations are done either when K is the field of rationals or the integers modulo a prime number.
A Fano curve is isomorphic to the projective line. A Fano surface is also called a del Pezzo surface. Every del Pezzo surface is isomorphic to either P 1 × P 1 or to the projective plane blown up in at most eight points, which must be in general position. As a result, they are all rational.