How can we know the Topology of $\{(x,y) \in \Bbb R ^2 : f(x,y)=0 \}$

Question

Let $f \in \Bbb R[x,y]$ be a given polynomial and set $$V=\{(x,y)\in \Bbb R^2 : f(x,y)=0\}$$

1. How can we tell if $V\neq \emptyset$ ?
2. How can we know if $V$ is compact?
3. How can we tell if $V$ is connected (or how many connected components it has)?

Of course, i don't take "ploting the set" as an answer. I want some kind of algorithm to answer this questions in terms of the given polynomial. Take in consideration that I'm working with real numbers, not complex.