I'm growing embarrassed by my lack of understanding for the sheaf-theoretic approaches to algebraic (and differential) geometry. I've tried to deal with it several times before, and I'm currently reading up on Algebraic Geometry again to fill the void that the finished thesis, soon arriving travels and non-existent job application responses produce.
So, why not learn by teaching? It's an approach that has been pretty darn good in the past. So I thought I'd write a sequence of posts on algebraic geometry, introducing what it's supposed to be about and how the main viewpoints develop more or less naturally from the approaches taken.
If we write our polynomials as coming from the ring [tex]k[x_1,\dots,x_n][/tex], then the corresponding solution points will be points in the vector space [tex]k^n[/tex]. In order to emphasize that we do not care for the vector space structure of this space, we shall denote it with [tex]\mathbb A^n[/tex], or if we want to emphasize the field, with [tex]\mathbb A^n_k[/tex].
The first observation at this point is that if we take the polynomial [tex]x^2+1[/tex], then the solution set over [tex]\mathbb R[/tex] is empty, while the solution set over [tex]\mathbb C[/tex] is not. So, in order to set all solution sets on an equal footing - and also to make the later occurring correspondences work out - we shall require [tex]k[/tex] to be an algebraically closed field. In other words, we can always find a root to any polynomial.
We call the solution sets varieties (or - in order to distinguish from everything else we might encounter, we shall call them affine algebraic varieties).
So, the study of solutions to systems of polynomial equations is the study of varieties. And hence geometry. This neatly expands on the classical linear algebra viewpoint - where we study systems of linear equations as intersections of planes. It turns out that the main computational approach - Gr