# Euler Characteristic of Real Algebraic Surfaces

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.

• That is a hard problem. Suppose $f=P(X)Q(Y)R(Z)$, where $P,Q,R$ are univariate polynomials that are pairwise coprime. Finding the Euler characteristic boils down to finding the number of real roots of each of these polynomials. Sep 18, 2015 at 20:18
• @LiviuNicolaescu Very interesting, could you just explain why the Euler characteristic boils down to finding the number of real roots of each of these polynomials? Sep 18, 2015 at 20:34
• I cannot fix the TeX typo in my comment above. Take for example the polynomial $$f=P(x)(y^2+1)(z^2+1).$$Then $V(f)$ is a union of planes, one plane for each real root of $P$. A formula for the Euler characteristic would lead to a formula forthe number of real roots. There are ways of determining this number, Hermite matrix, Sturm sequence, but for higher degrees they are not very practical. Sep 19, 2015 at 11:39