Let X be a germ of a complex variety, and R a ring of germs of locally bounded meromorphic functions on X. Prove that every element of R is a root of a monic polynomial over ${O}_X$.
I guess for an irreducible germ the locally bounded meromorphic function can only be holomorphic. But I am not sure how to do that, I think the definition, normal variety should be used to solve this problem.
A meromorphic locally bounded function on a non-normal variety need not be holomorphic. E.g., consider the function $\frac{z}{w}$. It is locally bounded on the cusp $z^2=w^3$ since on this variety $\left|\frac{z}{w}\right| = \frac{|w|^{3/2}}{|w|} = \sqrt{|w|}$. But this function is not holomorphic. On the other hand if we normalize the cusp by $t \mapsto (t^3,t^2)$, then the pullback via the normalization is (or extends to be) holomorphic: $\frac{z}{w} = \frac{t^3}{t^2} = t$.