Let $P\in \mathbb{R}[X,Y,Z]$ be an irreducible polynomial, such that $S=P^{-1}(0)$ is a non compact regular surface. If $K$ denotes the Gaussian curvature, prove that:$$\int_S |K|\leq 4\pi C(d)$$ Where $C(d)$ is a constant dependent on the degree of P.
If it were compact, one could take the maximum value of $|K|$ and use it to conjure an upper bound. Furthermore, if it were compact, Chern-Lashoff's Theorem would use the fact that the normal map is onto in order to gain a lower bound $\int_S |K| \geq 4 \pi$. However these techniques are all dependent on compactness, which means I really do not know how to estimate this integral.