Remarks on Artin approximation with constraints

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.

arXiv comment9 pages. To appear in Osaka J. Math