# Upper bound of multiplicity in prime characteristic

## TL;DR

This paper establishes an upper bound on the multiplicity of local rings in prime characteristic based on Frobenius test exponents, extending previous results and improving bounds for specific classes like F-nilpotent rings.

## Contribution

The paper provides a new upper bound on the multiplicity of local rings in prime characteristic using Frobenius test exponents, extending prior work and refining bounds for F-nilpotent rings.

## Key findings

- Derived an explicit upper bound for multiplicity involving Frobenius test exponent.
- Extended and improved bounds for F-nilpotent rings.
- Generalized previous results by Huneke, Watanabe, Katzman, and Zhang.

## Abstract

Let $(R, \frak m)$ be a local ring of prime characteristic $p$ of dimension $d$ with the embedding dimension $v$. Suppose the Frobenius test exponent for parameter ideals $Fte(R)$ of $R$ is finite, and let $Q = p^{Fte(R)}$. It is shown that $$e(R) \le Q^{v-d} \binom{v}{d}.$$ We also improve our bound for $F$-nilpotent rings. Our result extends the main results of Huneke and Watanabe and of Katzman and Zhang.