On The Linearity Of Artin Functions

TL;DR

This paper strengthens the understanding of Artin functions' linear bounds in excellent Henselian local rings, providing explicit calculations and analyzing the case of single-variable systems.

Contribution

It extends Elkik's linearity results to excellent Henselian rings and includes explicit computations for monomials and determinantal ideals.

Findings

Artin functions are linearly bounded in excellent Henselian rings.

Explicit formulas for Artin functions of monomials and determinantal ideals.

Linearity of Artin functions in one-variable polynomial systems.

Abstract

It was proved by Elkik that, under some smoothness conditions, the Artin functions of systems of polynomials over a Henselian pair are bounded above by linear functions. This paper gives a stronger form of this result for the class of excellent Henselian local rings. The linearity of Artin functions of systems of polynomials in one variable is also studied. Explicit calculations of Artin functions of monomials and determinantal ideals are also included.