I asked a question about details allowing to answer this question earlier today. Unfortunately, I didn't manage to complete the exercise. Since the other questions were about another problem, I write this new one.
Let $R$ be the ring of germs of $C^{\infty}$ functions on $\mathbb{R}^{n}$ in $0$. Let $K$ be the intersection of all powers of maximal ideal. I would like to prove that $K$ is generated by all nonnegative $\varphi\in R$.
We know there is only one maximal ideal:
$$M=\{f\in R\mid f(0)=0\}$$
whose powers are
$$M^{k}=\left\{f\in R\,\Big\vert\, \frac{\partial^{\vert\alpha\vert}f}{\partial x^{\alpha}}(0)=0,\,\forall\alpha\in\mathbb{N}^{n}:\vert\alpha\vert=0,1,\dots,k-1\right\}$$
We deduce that
$$K=\left\{f\in R\,\Big\vert\, \frac{\partial^{\vert\alpha\vert}f}{\partial x^{\alpha}}(0)=0,\,\forall\alpha\in\mathbb{N}^{n}\right\}$$
For any $A\in K$, we want to find a finite sequence of nonnegative functions in $K$, say $(\varphi_{i})_{i=1}^{m}$, and a finite sequence of functions in $R$, say $(f_{i})_{i=1}^{m}$, such that $A=\sum_{i=1}^{m}f_{i}\varphi_{i}$. Here is my attempt:
By Hadamard's lemma, we know that there exists a sequence of continuously differentiable functions $(g_{j})_{j=1}^{n}$ (the same $n$ as in $\mathbb{R}^{n}$) such that $A(x)=A(0)+\sum_{j=1}^{n}x_{j}g_{j}(x)$ for any $x\in\mathbb{R}^{n}$ where $x_{j}$ denotes the $j$-th component of $x$. I tried by decomposing $g_{j}$ in its positive part $\max\{0,g_{j}\}$ and negative part $\max\{0,-g_{j}\}$ but we still have problems with the coordinates...
The functions in $K$ are the so-called flat functions in $0$. An example in $\mathbb{R}$, where the function has at least one change in sign near $0$, is the following:
$$\begin{cases} e^{-1}+\sin(2e^{-1}(x-1)) &\text{if } x\geq 1\\ e^{-\frac{1}{x^{2}}} &\text{if } 0<x<1\\ 0 &\text{if } x=0\\ -e^{-\frac{1}{x^{2}}} &\text{if } -1<x<0\\ -e^{-1}+\sin(2e^{-1}(x+1)) &\text{if } x\le -1 \end{cases}$$
whose graph is represented in the following picture:
I did define the function piecewise to emphasize my intuition of the problem: we want to prove that locally, here in a neighbourhood of $0$, there exists a sequence of positive functions in $K$ such that the flat function can be expressed as a linear combination of this function with coefficients in the ring $R$ (not $\mathbb{R}$). My intuition is that, for a flat function, there exists a neighbourhood of $0$ such that this function has at most one change in sign in this neighbourhood, whatever the change in sign for "large" neighbourhoods of $0$. However, I can't prove it.
The idea behind this consideration is the following. If there exists a neighbourhood of $0$ such that the function has no change in sign in this neighbourhood, then the function respects the condition we want to prove. If there exists no neighbourhood of $0$ such that the function as at most one change in sign, then the function does "probably" (my intuition) has infinitely many changes in sign near $0$, which I believe to be impossible for such regular functions. However, these are heuristic yet not rigorous arguments, of course.
A solution would be to prove the following: for any real-valued function $A$, we can write $A(x) = \max\{A,0\}-\max\{-A,0\}$, and we would like to prove that if $A\in K$, then $\max\{A,0\}$ and $\max\{-A,0\}$ are in $K$.