Ideals and Varieties

In this past week, we have introduced a fair amount of new objects, so lets recap what we have done, as well as give an indication as to where we are headed.

We defined an affine variety V to be the set of points in affine n-space k^n satisfying a system of polynomial equations f_1 = ... = f_k = 0. There are many times in applications where one wants to understand properties of such a set. Varieties can describe many things, such as allowable positions of robotic arms, solutions to Sudoko puzzles, and others.

The description of an affine variety as a zero locus of a collection of polynomials is called an implicit description. When using such a description, it is easy to determine if a point in k^n is on the variety, but hard to find all the points. A description of (most of) the points on the variety via rational functions is called a parameterization of the variety. Parameterizations are nice because they make it easy to find points on the variety, but hard to know whether a given point is in the variety.

We did several examples of both polynomial and rational parameterizations of some different varieties, namely the circle, ellipse, parabola, and the tangent surface to the twisted cubic. One can always go from a rational parameterization of an affine variety to an implicit description using elimination theory, but not all varieties have a single rational parameterization of 'most' of the points. We'll see this later on in the class.

In linear algebra, one introduces the idea of a subspace in order to study solutions of linear systems. In order to study varieties, one introduces the notion of an ideal. An ideal is a subset of the polynomial ring that is closed under sums and "polynomial scalings", that is, if r is a polynomial, and f is in the ideal, then rf is also in the ideal. For us, we will be interested in the smallest ideal containing a set of polynomials f_1,...,f_k. This ideal can be thought of as the set of 'polynomial consequences' of the equations f_1 = 0, ..., f_k = 0. We'll spend a fair amount of time studying ideals, and algorithms that pertain to them, in order to study varieties.