Reduction of a family of ideals

TL;DR

This paper proves the existence of simultaneous reductions for a family of primary ideals in holomorphic function germs and generalizes a result on the semicontinuity of the Łojasiewicz exponent during deformations.

Contribution

It introduces a method for simultaneous reduction of ideals in holomorphic germs and extends previous semicontinuity results to broader classes of deformations.

Findings

Existence of simultaneous reductions for families of ideals.

Generalization of Łojasiewicz exponent semicontinuity.

Application to multiplicity-constant deformations.

Abstract

In the paper we prove that there exists a simultaneous reduction of one-parameter family of $\mathfrak{m}_{n}$-primary ideals in the ring of germs of holomorphic functions. As a corollary we generalize the result of A. P\l{}oski \cite{ploski} on the semicontinuity of the \L{}ojasiewicz exponent in a multiplicity-constant deformation.