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

Read more about this topic:  Algebraic Curve

Famous quotes containing the words plane, algebraic and/or curves:

    “Even though I had let them choose their own socks since babyhood, I was only beginning to learn to trust their adult judgment.. . . I had a sensation very much like the moment in an airplane when you realize that even if you stop holding the plane up by gripping the arms of your seat until your knuckles show white, the plane will stay up by itself. . . . To detach myself from my children . . . I had to achieve a condition which might be called loving objectivity.”
    —Anonymous Parent of Adult Children. Ourselves and Our Children, by Boston Women’s Health Book Collective, ch. 5 (1978)

    “I have no scheme about it,—no designs on men at all; and, if I had, my mode would be to tempt them with the fruit, and not with the manure. To what end do I lead a simple life at all, pray? That I may teach others to simplify their lives?—and so all our lives be simplified merely, like an algebraic formula? Or not, rather, that I may make use of the ground I have cleared, to live more worthily and profitably?”
    —Henry David Thoreau (1817–1862)

    “One way to do it might be by making the scenery penetrate the automobile. A polished black sedan was a good subject, especially if parked at the intersection of a tree-bordered street and one of those heavyish spring skies whose bloated gray clouds and amoeba-shaped blotches of blue seem more physical than the reticent elms and effusive pavement. Now break the body of the car into separate curves and panels; then put it together in terms of reflections.”
    —Vladimir Nabokov (1899–1977)