Lojasiewicz inequality over the ring of power series in two variables

TL;DR

This paper establishes a Lojasiewicz inequality for polynomial systems over power series rings in two variables, providing an effective version of the Strong Artin Approximation Theorem and bounds for Artin functions of isolated singularities.

Contribution

It introduces a Lojasiewicz inequality in the context of power series rings, extending the Strong Artin Approximation Theorem with explicit bounds.

Findings

Proves a Lojasiewicz inequality for power series coefficients

Derives bounds for Artin functions of isolated singularities

Provides an effective version of the Strong Artin Approximation Theorem

Abstract

We prove a Lojasiewicz type inequality for a system of polynomial equations with coefficients in the ring of formal power series in two variables. This result is an effective version of the Strong Artin Approximation Theorem. From this result we deduce a bound of Artin functions of isolated singularities.