Cubic curve

In Mathematical, a cubic curve is a plane Courbe defined by a cubic equation

F (X, Y, Z) = 0

in the homogeneous Coordinated of the projective plane ; or it is the not-homogeneous version for the Espace refines obtained by making Z = 1 in such an equation. Here F is a nonnull linear combination Monôme S of degree three

X ³, X ² Y ,…, Z ³

in X , Y and Z . Those are ten; thus the cubic curves form a projective Espace of dimension 9, above any body K given. Each point P imposes only one linear condition on F , if we ask C to pass by P . Thus we can find a curve cubic passing through any family of nine points given in advance.

If one seeks the cubic ones which passes by $8 \,$ points given, one obtains for the coefficients of his equation a homogeneous linear system of $8 \,$ equations with $10 \,$ unknown. If these points are " in position générale" (the algebraic Geometry is made inter alia including/understanding what that wants to say!) the row of this system is maximum $8 \,$ .

If $C_1=0 \,$ and $C_2=0 \,$ are the equations of two among them, the others of form $\ lambda_1C_1+ \ lambda_2C_2=0 \,$ . They pass all by the points of intersection of these two cubic. THERE is $9 \,$ according to Theorem of Bezout.

We have just shown, in a a little light way it is true, that all the cubic plane ones which passes by $8 \,$ points " in position générale" pass by a $9 \,$ ième point. This result is in particular used to prove the Associativeness of the law of group defined on cubic the nonsingular ones, to see the elliptic article Curve.

A cubic curve can have a singular Point; in this case it has a parameterization by a projective Droite. If not a cubic curve nonsingular is known to have nine points of inflection above a Corps algebraically closed such as the complex numbers. That can be shown by taking the homogeneous version of the definite matrix hessienne cubic, and by intersecting its determinant with C; the intersections are then counted to the Théorème of Bézout. These points cannot however be all real, so that they cannot be seen in the real projective plan by plotting the curve. The real points of the cubic curves were studied by Newton; they form one or two oval .

Cubic a nonsingular defines a elliptic Courbe, on any body K for which it has a point with coordinates in K (not K - rational). The elliptic curves on the body of the complex numbers are now often studied by using the elliptic Fonctions of Weierstrass. These elliptic functions (for a given network) form an isomorphous body with the body of the rational functions of cubic of equation refines $y^2 = X (x-1) (X \ lambda)$ . The possibility for cubic on K of having such a form of Weierstrass depends on the existence on a point K - rational, which is used like Point ad infinitum in the form as Weierstrass. For example, there are several cubic curves which do not have such a point, when K is the body of the rational numbers.

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