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I think you should consider Pietro's comment. What I meant to illustrate is that you probably have a non-working definition: for $k=1$, say, the derivative you compute is what I would call the derivative of $f_n$ at $0$ in the direction $x$.
If $1$ and $0$ are not in the same connected component of $\Omega$ (in particular if $x$ and $0$ lie in different components of $U$), you can define $f_n=0$ near $0$ and $f_n=n$ near $x$. In that case I fail to see why there should exist a link between what happens near $x$ and near $0$.
@user64494 ok, but do you have a proof that Mathematica's code cannot fail? Are you sure your plot does not contain any small-scale distorsion? There are a lot of questions to answer before a Mathematica's output can be considererd sound mathematics (I'm not saying it's not possible but it requires additional work)
@Ali: let me kindly disagree with your last comment ;) I simply don't know how to characterize tangent-to-identity real-analytic germs which are locally analytically equivalent to holomorphic parabolic germs in terms of e.g. their Taylor coefficients or their functional properties (like solving ODE/PDE for instance) .
In the case $n=1$ how do you know that $f$ is not surjective? Does it stem from the null-homotopic property, or do you assume some higher regularity on $f$ than mere continuity?
If I understand correctly, $N$ is given and you look for $M$. This seems to be related to the Frobenius theorem. Is this the kind of result you have in mind? In that special setting you have obstructions that are encoded by an integrability condition, imposing restrictions on $N$. (There also exists infinte-dimensional versions, e.g. for Banach manifolds.)
It looks like you are considering degree-2 algebraic systems, but (up to add dummy variables) any algebraic system has degree 2. So (for general $K$) it shouldn't be easier to solve than your favorite arbitrary system. Or am I missing something (I'm more used to complete char-0 fields, so I probably am) ?
@MichaelHardy: I meant that some people try to avoid using the operator-like notation $\sin x$ in favor of the evaluation form $\sin(x)$. Not so much a matter of ambiguity as notational preferences, I'd say. For instance, for teaching material I always use the evaluation form (the operator notation leads more easily to multiplicative simplifications or other nonsense that I hope to avoid this way), but I'm more relaxed in research papers.
