Algebraic Curve

A algebraic curve is a Courbe , generally planes , whose Cartesian equation can put in polynomial form . An algebraic curve not is known as transcendent .

In algebraic geometry, a curve is a algebraic Variété whose related component are very of dimension 1. In practice, one often restricts oneself with the not-singular and related projective curves.

Definition

A algebraic curve is more formally the whole of the points of a geometrical Espace whose Cartesian Coordonnées are solutions of a algebraic equation .

The geometrical Espace considered is generally the plane closely connected Euclidean reality , but it is possible:

- to resort to spaces of size higher than two (space or hyperspace instead of the plan);

- to place itself within the framework of an other geometry which refines Euclidean (projective for example);

- to work with another basic body that of realities (in Cryptography for example, one uses plans on Corps finished S ).

We will limit ourselves however here to the case of the plane refines Euclidean reality .

The Cartesian Coordonnées of a point M in the plan are two numbers (usually real, but that can depend on the plan considered) respectively called ordered , and usually noted X-coordinate and X and there . They indicate the values of projections of the point M on two orthogonal axes of the plan.

A algebraic equation in the plan is an equation which can be put in the form:

$P (X, there) = 0 \,$
where P ( X , there ) indicates a irreducible Polynôme of degree not no one of the Cartesian coordinates X and there .

Order and classification
The polynomial P thus associated with a curve is not single; in fact, it is defined near only with one multiplicative constant: if P is associated with a curve, then any polynomial λ. P where λ is real not no one is also associated for him. However, all these polynomials are same degree, called order of the algebraic curve.
The algebraic curved can thus be classified according to their order N :

for N = 1 , we have the rectic ; it is in fact the right ;

for N = 2 , we have the Conique S , thus called because it is possible to obtain them like intersection of a cone and a plan; they are divided into three families:

- the ellipses , of which the circle ;
- the hyperboles , of which the hyperbole équilatère ;
- and the parabolas .

for N = 3 , we have the cubic ;

for N = 4 , we have the quartics ;

for N = 5 , we have the quintic ;

for N = 6 , we have the sextic ;

for N = 7 , we have the septic ;

for N = 8 , we have the octic , or biquartic ;

for N = 9 , we have the nonic , or tricubic ;

for N = 10 , we have the decic , or biquintic ;

for N = 11 , we have the undecic ;

for N = 12 , we have the duodecic , or triquartic ;

beyond, and even rather often starting from order 9, one rather speaks about “ algebraic curve of order N ”…

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