# Uniform Symbolic Topologies in Normal Toric Rings

## TL;DR

This paper establishes a uniform bound for symbolic powers in normal toric rings, explicitly computes this bound using polyhedral data, and links it to the F-signature in positive characteristic.

## Contribution

It provides an explicit computation of the uniform integer D for symbolic power containment in normal toric rings, connecting algebraic and geometric data.

## Key findings

- Computed a uniform integer D for symbolic power containment
- Explicit formula for D in terms of polyhedral cone data
- Linked the uniform bound to the F-signature in positive characteristic

## Abstract

Given a normal toric algebra $R$, we compute a uniform integer $D = D(R) > 0$ such that the symbolic power $P^{(D N)} \subseteq P^N$ for all $N >0$ and all monomial primes $P$. We compute the multiplier $D$ explicitly in terms of the polyhedral cone data defining $R$. In this toric setting, we draw a connection with the F-signature of $R$ in positive characteristic.