Given a (compact if needed) real smooth surface $V(f)$ defined by $f\in \mathbb{R}[X,Y,Z]$, in particular it is oriented. Is there a formula which gives the Euler character of $V(f)$ ?


Thanks. 



Euler Characteristic of Real Algebraic Surfaces
A “formula” is a lot to ask for, but there are *algorithms* based on Morse theory.


E.g. §5 of [Basu (1999)](http://www.ams.org/mathscinet-getitem?mr=1692627), or §3 of [Fortuna-Gianni-Luminati (2004)](http://www.ams.org/mathscinet-getitem?mr=2169368).


