Zariski-Nagata Theorems for Singularities and the Uniform Izumi-Rees Property

We introduce and explore the Uniform Izumi-Rees Property in Noetherian rings with applications to multiplicity theory and containment relationships among symbolic powers of ideals. As an application, we prove that if R is a normal domain essentially of finite type over a field, there exists a constant C so that for all prime ideals p⊆ q(R), if p⊆ q⁽ᵗ⁾, then for all n, there is a containment of symbolic powers p(Cn)⊆ q⁽ᵗⁿ⁾.