Simple closed planar algebraic curves What are simple closed planar algebraic curves? Is any classification known? Of course, I know that the ellipse with the equation $ax^2+by^2-1=0$  is such a closed curve. I can imagine something similar in the case of even degree, e.g. $ax^4+by^4-1=0$. Are there any simple closed algebraic curves of odd degree?
 A: Answer to your initial question : First, as Arthur noticed, any curve of odd degree is unbounded so it's not closed simple curve. On the other hand, for "trivial" reasons, for any degree $2k$ there is a simple closed curve of degree $2k$ : take several empty conics (i.e curves with no points, for example $x^2 + y^2 = -1$) and take a non-empty conic + $k-1$ empty conics. For example, for $k = 3$ you obtain the simple closed curve $(x^2 + y^2 -1)(x^2 + y^2 + 1)(x^2 + y^2 +3) = 0$. You can even obtain irreducible polynomials by modifying a little the constant term.
Some remarks : On the other hand, it is more interesting to look at smooth plane algebraic curves, since usually a curve has (many!) connected component, for example the elliptic curve $y^2 = x(x-1)(x-2)$ has two components, one bounded and one not bounded. Indeed, one has the famous bound of Harnack, saying that in the real projective plane, a curve of degree $d$ has no more that $\frac{(d-1)(d-2)}{2} + 1$ connected component. It is of course false for $\Bbb R^2$ as an hyperbola has $2$ connected components. This is a fascinating subject, and the website of O. Viro has numerous informations about topology of algebraic curves. Of course, one could also try to classify them up to more "rigid" transformations but it is very hard.
Edit (Example) : I did realized you wanted example of different degree. 
For degree $3$, there is plenty of curves already (it was already studied by Newton indeed). The curve $y^2 = x^3$ is the famous "cusp", with a singularity at the origin. This is the intermediate state between the connected curve $y^2 = x(x^2 + 1)$ and the curve $y^2 = x(x-1)(x+1)$ which contains two connected components. For degree $4$, there are plenty of examples. Since it grows exponentially, best way would be to learn how to perturb curves since it is a good way of creating new curves from old one.
Perturbation of curves The idea is to take the union of two curves, and then slightly modify this union for get an irreducible smooth curve. Let's take for example the circle and the line $y = 0$. We obtain the equation $(x^2 + y^2 - 1)y = 0$. What happens if we take instead the equation $(x^2 + y^2 -1)y = t$ for $t > 0$ small ? Well, the zero set of $(x^2 + y^2 - 1)y$ separes $\Bbb R^2$ into positive zones and negatives zones. Basically, this $t$ will shrink the positive zone and make bigger the negative zone. At a point of intersection of the circle and the line, you will get locally a "cross" with 2 + and 2 - sign regions. So perturbing with this $t$ will simply put together the $-$ regions and modify the curve ! Using this you can easily classify curves of degree $\leq 5$ topologically. Alreay for $6$ it's non trivial. 
