Geometry – When is a plane algebraic curve closed?

When is a flat algebraic curve closed?

Is there a metric? For square http://mathworld.wolfram.com/QuadraticCurveDiscriminant.html

We know that the discriminant must be below 0.

Is there something for higher degrees? I'm not looking for suggestions. I am looking for complete answers. No proofs of existence, because I know they exist.

Do we know it and what is it?