= Solution
A sufficient condition is that the <derivative> $Df_x:\mathbb R^n\to\mathbb R^m$ is <surjective>, equivalently that its <Jacobian matrix> has rank $m$. Then the <implicit function theorem> makes the zero set near $x$ a $C^1$ <embedded submanifold> of dimension
$$
\boxed{n-m.}
$$
The rank condition persists nearby because the derivative is continuous. This is the local form of the <regular level set theorem>; only the rank at the specified point is needed for this local conclusion.
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