Algebraic curve
In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables.
These two operations are each inverse to the other; therefore, the phrase algebraic plane curve is often used without specifying explicitly whether it is the affine or the projective case that is considered.
An algebraic curve in the Euclidean plane is the set of the points whose coordinates are the solutions of a bivariate polynomial equation p(x, y) = 0.
These problems are not as easy to solve as in the case of the graph of a function, for which y may easily be computed for various values of x.
The fact that the defining equation is a polynomial implies that the curve has some structural properties that may help in solving these problems.
To draw an algebraic curve, it is important to know the remarkable points and their tangents, the infinite branches and their asymptotes (if any) and the way in which the arcs connect them.
Intersecting with a line parallel to the axes allows one to find at least a point in each branch of the curve.
If an efficient root-finding algorithm is available, this allows to draw the curve by plotting the intersection point with all the lines parallel to the y-axis and passing through each pixel on the x-axis.
In the case of polynomials, another formula for the tangent has a simpler constant term and is more symmetric:
The equivalence of the two equations results from Euler's homogeneous function theorem applied to P. If
The general formula for a tangent to a projective curve may apply, but it is worth to make it explicit in this case.
be the decomposition of the polynomial defining the curve into its homogeneous parts, where pi is the sum of the monomials of p of degree i.
This implies that the number of singular points is finite as long as p(x,y) or P(x,y,z) is square free.
Bézout's theorem implies thus that the number of singular points is at most (d − 1)2, but this bound is not sharp because the system of equations is overdetermined.
The maximum is reached by the curves of genus zero whose all singularities have multiplicity two and distinct tangents (see below).
This is a corollary of the analytic implicit function theorem, and implies that the curve is smooth near the point.
Near a singular point, the situation is more complicated and involves Puiseux series, which provide analytic parametric equations of the branches.
These factors are all different if f is an irreducible polynomial, because this implies that f is square-free, a property which is independent of the field of coefficients.
The algebraic curve corresponding to the function field is simply the set of points (x, y) in C2 satisfying y2 = x3 − x − 1.
Using the intrinsic concept of tangent space, points P on an algebraic curve C are classified as smooth (synonymous: non-singular), or else singular.
Since f is a polynomial, this definition is purely algebraic and makes no assumption about the nature of the field F, which in particular need not be the real or complex numbers.
In the remainder of this section, one considers a plane curve C defined as the zero set of a bivariate polynomial f(x, y).
The multiplicity m is defined as the maximum integer such that the derivatives of f to all orders up to m – 1 vanish (also the minimal intersection number between the curve and a straight line at P).
[1] The Milnor number μ of a singularity is the degree of the mapping grad f(x,y)/|grad f(x,y)| on the small sphere of radius ε, in the sense of the topological degree of a continuous mapping, where grad f is the (complex) gradient vector field of f. It is related to δ and r by the Milnor–Jung formula, Here, the branching number r of P is the number of locally irreducible branches at P. For example, r = 1 at an ordinary cusp, and r = 2 at an ordinary double point.
Computing the delta invariants of all of the singularities allows the genus g of the curve to be determined; if d is the degree, then
Concretely, a rational curve embedded in an affine space of dimension n over F can be parameterized (except for isolated exceptional points) by means of n rational functions of a single parameter t; by reducing these rational functions to the same denominator, the n+1 resulting polynomials define a polynomial parametrization of the projective completion of the curve in the projective space.
Drawing a line with slope t from (−1,0), y = t(x + 1), substituting it in the equation of the ellipse, factoring, and solving for x, we obtain
All points of the ellipse are given, except for (−1,1), which corresponds to t = ∞; the entire curve is parameterized therefore by the real projective line.
If the characteristic of the field is different from 2 and 3, then a linear change of coordinates allows putting
In a plane cubic model three points sum to zero in the group if and only if they are collinear.
