A division's theorem on some class of $\mathcal{C}^\infty$-functions

TL;DR

This paper extends the division theorem for ideals in the ring of smooth function germs, showing that ideals generated by finitely many quasi-analytic functions are closed, unlike the general case.

Contribution

It proves a division theorem for ideals generated by finitely many quasi-analytic germs in the ring of smooth functions, broadening known results beyond analytic functions.

Findings

Ideals generated by finitely many analytic germs are closed.

The division theorem holds for certain quasi-analytic function classes.

Counterexamples exist for general ideals of finite type.

Abstract

Let $\mathcal{E}_n$ be the ring of the germs of $\mathcal{C}^\infty$-functions at the origin in $\R^n$. It is well known that if $I$ is an ideal of $\mathcal{E}_n$, generated by a finite number of germs of analytic functions, then $I$ is closed. In this paper we consider an ideal of $\mathcal{E}_n$ generated by a finite number of germs in some class of $\mathcal{C}^\infty$-functions that are not analytic in $\`a$, but quasi-analytic and we shall prove that the result holds in this general situation. We remark that the result is not true for a general ideal of finite type of $\mathcal{E}_n$.