Remarks on Artin approximation with constraints

TL;DR

This paper investigates approximation solutions to equations with constraints on variable dependencies, extending Artin's work to cases where solutions do not depend on all variables, with implications for algebraic geometry.

Contribution

It provides new approximation results for solutions of equations with variable dependency constraints, expanding Artin's classical approximation theory.

Findings

New approximation results for constrained solutions

Extension of Artin's approximation to partial variable dependence

Implications for algebraic geometry and solution structures

Abstract

We study various approximation results of solutions of equations $f(x,Y)=0$ where $f(x,Y)\in\mathbb K[[x]][Y]^r$ and $x$ and $Y$ are two sets of variables, and where some components of the solutions $y(x)\in\mathbb K[[x]]^m$ do not depend on all the variables $x_j$. These problems have been highlighted by M. Artin.