Suppose $X$ is an irreducible affine variety (i.e. embedded in some affine space) with coordinate ring $\Gamma(X)$. Suppose $\phi$ is a nonzero rational function on $X$. Let $p$ be a point on $X$. Is it true that $\phi$ has a representative as a ratio of two polynomials (or two elements in $\Gamma(X)$) $f/g$ such that we do not have $f(p) = 0$ and $g(p) = 0$ at the same time, i.e. we can find a representative of $\phi$ so that it is unambiguous whether or not $p$ is a pole?
Another way to phrase the question. Let $A = \{f \in \Gamma(X) : f \cdot \phi \in \Gamma(X)\}$ (the ideal generated by denominators of possible representatives) and let $B = \{f \in \Gamma(X) : f \cdot \frac{1}{\phi} \in \Gamma(X)\}$ (ideal generated by numerators of possible representatives). Is it true that $A + B = \Gamma(X)$ i.e. the common vanishing locus of $A$ and $B$ is the empty set?
You can assume whatever nice properties you want, i.e. you can assume we're working over $\mathbb{C}$.