If I have a (bijective) scalar function $f : \mathbb{R}^N \to \mathbb{R}$ and $\vec{x}\in \mathbb{R}^N$. Is it possible to obtain an inverse function $h:\mathbb{R} \to \mathbb{R}^N $ such that $\vec{h}(f(\vec{x}))=\vec{x}$? A series expansion would suffice.
I believe that a multivariate extension of Lagrange Inversion Theorem (eg. see equation (1) in this paper) should yield the answer but I cannot see how it would be possible to get a vector valued output.
Related but different questions: Inverse of multivariate function power series, Existence of an inverse of a multivariate function, Multivariate Lagrange inversion with powers
