# Uniform Symbolic Topologies via Multinomial Expansions

## TL;DR

This paper investigates conditions under which symbolic powers of prime ideals are contained within ordinary powers in non-regular rings, providing multinomial expansion criteria for such containments in tensor products of domains.

## Contribution

It introduces multinomial expansion criteria for symbolic power containments in tensor products of domains, extending known results beyond regular rings.

## Key findings

- Provides binomial and multinomial criteria for ideal containments.
- Establishes conditions for uniform symbolic topologies in non-regular rings.
- Includes an example illustrating the application of these criteria.

## Abstract

When does a Noetherian commutative ring $R$ have uniform symbolic topologies on primes--read, when does there exist an integer $D>0$ such that the symbolic power $P^{(Dr)} \subseteq P^r$ for all prime ideals $P \subseteq R$ and all $r >0$? Groundbreaking work of Ein-Lazarsfeld-Smith, as extended by Hochster and Huneke, and by Ma and Schwede in turn, provides a beautiful answer in the setting of finite-dimensional excellent regular rings. It is natural to then sleuth for analogues where the ring $R$ is non-regular, or where the above ideal containments can be improved using a linear function whose growth rate is slower. This manuscript falls under the overlap of these research directions. Working with a prescribed type of prime ideal $Q$ inside of tensor products of domains of finite type over an algebraically closed field $\mathbb{F}$, we present binomial- and multinomial expansion criteria for containments of type $Q^{(E r)} \subseteq Q^r$, or even better, of type $Q^{(E (r-1)+1)} \subseteq Q^r$ for all $r>0$. The final section consolidates remarks on how often we can utilize these criteria, presenting an example.