I just started to read Smith's Invitation to Algebraic Geometry. On one of the first pages it states:
An affine plane curve is the zero set of one complex polynomial in the complex plane.
By definition, an algebraic variety is the zero locus of a collection of polynomials.
For the purpose of this question let's assume the affine space is $\mathbb C^n$ or $\mathbb R^n$.
It is not clear to me why the dimension of the variety should equal the number of polynomials and whether or not the degrees matter.
(For example, the zero locus of two polynomials that produce two unit circles touching at the origin seems to me is a curve also. What am I missing?) (<-- this is a bad example that I came up with because I don't yet understand the subject well.)
Please could someone help me understand why an affine algebraic variety has dimension one if and only if it is the zero locus of exactly one polynomial of two variables (this is how I interpret what the book states)?
Here, instead of a bad example, perhaps it's better if I ask it as a question:
Is it impossible to start with an affine space, say $\mathbb R^3$ or $\mathbb R^4$ and then produce two polynomials so that their zero locus is a curve?