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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 ...
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
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 ]
The Birch and Swinnerton-Dyer conjecture on elliptic curves postulates a connection between the rank of an elliptic curve and the order of pole of its Hasse–Weil L-function. It has been an important landmark in Diophantine geometry since the mid-1960s, with results such as the Coates–Wiles theorem , Gross–Zagier theorem and Kolyvagin's ...
The above solution shows that a quartic polynomial with rational coefficients and a zero coefficient on the cubic term is factorable into quadratics with rational coefficients if and only if either the resolvent cubic has a non-zero root which is the square of a rational, or p 2 − 4r is the square of rational and q = 0; this can readily be ...
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.
This shows that an algebraic number can be equivalently defined as a root of a polynomial with either integer or rational coefficients. Given an algebraic number, there is a unique monic polynomial with rational coefficients of least degree that has the number as a root. This polynomial is called its minimal polynomial.