Find conditions in which the function $F$ has a continuously differentiable inverse function

Question

Let $f\colon\mathbb C\to\mathbb C$ be an entire non constant function. We consider its values on the real line. The function $f$ has infinitely many real zeros and there is infinitely many real solutions to the equation $f(s)=w$ for any real number $w$. Here the set of those real $s$ is a discrete set. The same properties holds true for all the $k^{th}$ derivatives of $f$.

Let $$F:U⊂ℝ^{r+1}→ℝ^{r+1}$$ $$(s₁,s₂,...,s_{r},s_{r+1})→F(s₁,s₂,...,s_{r},s_{r+1})=(f′(s₁),f′′(s₂),f′′′(s₃),.....,f^{(r)}(s_{r}),f^{(r+1)}(s_{r+1}))$$ be a continuously differentiable function defined from an open set $U⊂ℝ^{r+1}$ into $ℝ^{r+1}$.

My question is : Find sufficient and necessary conditions in which the function $F$ has a continuously differentiable inverse function defined in some neighborhoods

Answer 1

The Jacobian of $F$ is $$ \prod_{k=1}^{r+1}f^{(i+1)}(s_i). $$ A sufficient condition for the existence of $F^{-1}$ in a neighborhood of $(s_1,\dots,s_{r+1})$ is $f^{(i+1)}(s_i)\ne0$, $1\le i\le r+1$. Since $$ F^{-1}(t_1,\dots,t_{r+1})=\bigl((f^{(i)})^{-1}(t_i)\bigr)_{i=1}^{r+1}, $$ it is also necessary for $F^{-1}$ to be $C^1$.