Limits of length functions of multi $p$-families of ideals

TL;DR

This paper investigates the asymptotic behavior of length functions of multi-$p$-families of ideals in characteristic $p$, establishing relationships with shifted families and extending known formulas.

Contribution

It introduces a generalized asymptotic relationship between length functions of multi-$p$-families and their shifted counterparts under linear growth conditions.

Findings

Established asymptotic relationships for length functions in multi-$p$-families.

Extended Wantanabe-Yoshida formula to certain $p$-families.

Provided examples including products of Frobenius powers and mixed multiplicities.

Abstract

We show the asymptotic relationship between the limit of the normalized length function of a multi-$p-$family of ideals and that of its shifted family under linear growth conditions in a local domain of characteristic $p$. Examples of multi-$p-$families of ideals including products of Frobenius powers of different ideals. We apply our results to obtain a generalized version of a formula due to Wantanabe-Yoshida for certain $p-$families using results from Verma, and to provide an instance of the existence of a mixed multiplicity version of multi-$p$-families of ideals.