Number of connected components of complement of plane curve

Question

Let $$P(x,y)=\sum_{k=0}^m\sum_{l=0}^n a_{k,l}x^k y^l$$ be a polynomial over $\mathbb R$ in $x$ and $y$ of degree $m+n$. The zero locus $$\mathcal C=\{(x,y)\in \mathbb R^2\,|\, P(x,y)=0\}$$ is a plane algebraic curve. How many connected components does $\mathbb R^2\backslash \mathcal C$ have?