Existence of uniform symbolic multipliers in singular domains not containing a field

Determine whether a broad class of Noetherian singular domains that do not contain a field—such as domains essentially of finite type over the integers—admits a uniform symbolic multiplier; specifically, ascertain the existence of a nonzero element z in R and a constant C in the natural numbers such that for every ideal I subset of R and every n in the natural numbers, the containment z^n · I^{(C n)} ⊆ I^n holds. Additionally, investigate whether for any element c ≠ 0 with the localization R_c regular, some power of c serves as a uniform symbolic multiplier in the same sense.

Background

Uniform symbolic multipliers are elements that uniformly convert symbolic powers of all ideals into linearly bounded ordinary powers. In equicharacteristic settings, such multipliers can be obtained using Jacobian-ideal techniques and Frobenius methods. These play a key role in proving uniform containments I^{(Cn)} ⊆ I^n and thus in establishing the Uniform Symbolic Topology Property.

For mixed characteristic rings (or more generally rings not containing a field), the existence of uniform symbolic multipliers is central yet unresolved. The paper highlights that many advances on the Uniform Symbolic Topology Property in equicharacteristic rely on such multipliers together with the Uniform Artin–Rees and Uniform Briançon–Skoda Theorems, and proposes focusing on rings essentially of finite type over ℤ, which enjoy these uniform properties. The authors then formulate concrete questions about the existence of such multipliers and whether powers of elements whose localizations are regular can serve as uniform symbolic multipliers.