So, if you have an irreducible variety $X$ over $k=\bar{k}$ and you consider a rational function $\phi=f/g\in k(X)$, is there a general way to determine where $\phi$ is not regular?
At this point I would like to avoid sheaf-theoretic stuff, as long as it is possible.
What I know
If you only have one point where $f$ could not be regular then you only have to prove that $f\notin k[X]$. This is what happens with $X=V(x^2+y^2−1)$ and $(y-1)/x$ here: when is rational function regular? However, in that particular case it is simple and you have a trick adapted.
If you suspect that $f$ could not be regular at the hypersurface $Y\subset X$ at some regular points of $X$ then you can use the local equation $\pi$ of $Y$ in certain open set to compute the number $M_g$ such that $g\in(\pi^{M_g})$ ($M_g$ max., $M_f$ similar). So $\phi$ is regular iff $M_f-M_g\geq0$. These $M$ are usually denoted $v_Y(g)$.
What I don't know
What happens in general? In particular: What happens with the singular points of $X$? And if $Y$ has greater codimension?