Bornes effectives des fonctions d'approximation des solutions formelles d'\'equations binomiales

TL;DR

This paper develops effective bounds for the solutions of binomial equations using an improved Artin Approximation Theorem, showing doubly exponential bounds generally and affine bounds under certain conditions.

Contribution

It provides the first effective bounds for the Artin function specifically for binomial equations, extending classical approximation results.

Findings

Artin function for binomial equations is bounded by a doubly exponential function

Bounded by an affine function when the order of solutions is limited

Effective version of Greenberg Approximation Theorem applied to binomial systems

Abstract

The aim of this paper is to give an effective version of the Strong Artin Approximation Theorem for binomial equations. First we give an effective version of the Greenberg Approximation Theorem for polynomial equations, then using the Weierstrass Preparation Theorem, we apply this effective result to binomial equations. We prove that the Artin function of a system of binomial equations is bounded by a doubly exponential function in general and that it is bounded by an affine function if the order of the approximated solutions is bounded.