Approximation diophantienne dans les corps de series en plusieurs variables

TL;DR

This paper studies Diophantine approximation in rings of multivariable power series and establishes bounds on the Artin function for homogeneous polynomials, linking it to Lojasiewicz inequalities.

Contribution

It introduces new bounds for the Artin function in multivariable power series rings, connecting Diophantine approximation with valuation theory and inequalities.

Findings

Bound on the Artin function for homogeneous polynomials in two variables

Connection between Diophantine approximation and Lojasiewicz inequalities

Results applicable to valuation theory in power series rings

Abstract

We give here a result of diophantine approximation between $\O_N$, the ring of power series in several variables, and the completion of the valuation ring that dominates $\O_N$ for the $\m$-adic topology. We deduce from this that the Artin function of a homogenous polynomial in two variables is bounded by an affine function, which can be interpreted in term of a Lojasiewicz inequality on the wedges space.