New estimates of Hilbert-Kunz multiplicities for local rings of fixed dimension

TL;DR

This paper advances the understanding of Hilbert-Kunz multiplicities in local rings by proving the Watanabe-Yoshida conjecture for rings with low multiplicity or dimension, and introduces new bounds for non-regular rings.

Contribution

It improves volume estimates to confirm the Watanabe-Yoshida conjecture in specific cases and establishes new lower bounds for non-regular rings of fixed dimension.

Findings

Conjecture verified for rings with Hilbert-Samuel multiplicity ≤ 5.

Conjecture verified for rings of dimension ≤ 6.

New lower bounds for Hilbert-Kunz multiplicity in non-regular rings.

Abstract

We present results on the Watanabe-Yoshida conjecture for the Hilbert-Kunz multiplicity of a local ring of positive characteristic. By improving on a "volume estimate" giving a lower bound for Hilbert-Kunz multiplicity, we obtain the conjecture when the ring either has Hilbert-Samuel multiplicity less than or equal to five, or dimension less than or equal to six. For non-regular rings with fixed dimension, a new lower bound for the Hilbert-Kunz multiplicity is obtained.