Algebraic Curve - Non Plane Algebraic Curves

Non Plane Algebraic Curves

An algebraic curve is an algebraic variety of dimension one. This implies that an affine curve in an affine space of dimension n is defined by, at least, n -1 polynomials in n variables. To define a curve, these polynomials must generate a prime ideal of Krull dimension 1. This condition is not easy to test in practice. Therefore the following way to represent non plane curves may be preferred.

Let be n-1 polynomials in two variables and such that f is irreducible. The points in the affine space of dimension n such whose coordinates satisfy the equations and inequations

$\begin{align} &f(x_1,x_2)=0\\ &g_0(x_1,x_2)\neq 0\\ x_3&=\frac{g_3(x_1,x_2)}{g_0(x_1,x_2)}\\ \vdots &\\ x_n&=\frac{g_n(x_1,x_2)}{g_0(x_1,x_2)} \end{align}$

are all the points of an algebraic curve in which a finite number of points have been removed. This curve is defined by a system of generators of the ideal of the polynomials h such that it exists an integer k such belongs to the ideal generated by .

This representation is a rational equivalence between the curve and the plane curve defined by f. Every algebraic curve may be represented in this way. However, a linear change of variables may be needed in order to make almost always injective the projection on the two first variables. When a change of variables is needed, almost every change is convenient, as soon as it is defined over an infinite field.

This representation allows to deduce easily any property of a non-plane algebraic curve, including its graphical representation, from the corresponding property of its plane projection.

For a curve defined by its implicit equations, above representation of the curve may easily deduced from a Gröbner basis for a block ordering such that the block of the smaller variables is The polynomial f is the unique polynomial in the base that depends only of and . The fractions are obtained by choosing, for i = 3, ..., n, a polynomial in the basis that is linear in and depends only on and . If these choices are not possible, this means either that the equations define an algebraic set that is not a variety, or that the variety is not of dimension one, or that one must change of coordinates. The latter case occurs when f exists and is unique, and, for i = 3, ..., n, there exist polynomials whose leading monomial depends only on and

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