Asymptotic behaviour of standard bases

TL;DR

This paper investigates the growth of orders of elements in standard bases of powers of ideals in Noetherian local rings and power series rings, establishing linear and polynomial bounds on their orders.

Contribution

It proves that elements of standard bases of $I^n$ have orders bounded linearly in $n$ in Noetherian local rings and polynomially in $n$ in power series rings, advancing understanding of their asymptotic behavior.

Findings

Orders are linearly bounded in Noetherian local rings.

Orders are polynomially bounded in power series rings.

Provides new bounds on standard bases of ideal powers.

Abstract

We prove here that the elements of any standard basis of $I^n$, where $I$ is an ideal of a Noetherian local ring and $n$ is a positive integer, have order bounded by a linear function in $n$. We deduce from this that the elements of any standard basis of $I^n$ in the sense of Grauert-Hironaka, where $I$ is an ideal of the ring of power series, have order bounded by a polynomial function in $n$.