Let $G_1, ..., G_s : \mathbb{R}^s \to \mathbb{R}^s$ be diffeomorphisms on an open unit ball $B \subset \mathbb{R}^s$, centered at $0$, onto their images. Let $m_1, .., m_s$ be integers not all $0$, and let $F = m_1 G_1 + ... + m_s G_s$. Suppose we know that the determinant of the Jacobian matrix of $F$ at $x$ is non-zero for all $x \in B$.
Does it then follow that $F$ is still a diffeomorphism on $B$?
If $F$ is not a diffeomorphism on $B$, is it possible to show that $$ \# F^{-1}(y) < C $$ for all $y$ in $F(B)$ for some positive $C$?
Any comments are appreciated!
Ps edits had been made based on comments