Surjectivity of quotient maps for algebraic $(\mathbb{C},+)$-actions and polynomial maps with contractible fibres

In this paper, we establish two results concerning algebraic $(\mathbb{C},+)$-actions on ${\mathbb{C}}^n$. First let $\varphi$ be an algebraic $(\mathbb{C},+)$-action on ${\mathbb{C}}^3$. By a result of Miyanishi, its ring of invariants is isomorphic to $\mathbb{C}[t_1,t_2]$. If $f_1,f_2$ generate this ring, the quotient map of $\varphi$ is the map $F:$\mathbb{C}$^3\to $\mathbb{C}$^2$,$x\mapsto (f_1(x),f_2(x))$. By using some topological arguments, we prove that $F$ is always surjective. Second, we are interested in dominant polynomial maps $F:{\mathbb{C}}^n\to {\mathbb{C}}^{n-1}$ whose connected components of their connected fibres are contractible. For such maps, we prove the existence of an algebraic $(\mathbb{C},+)$-action $\varphi$ on ${\mathbb{C}}^n$ for which $F$ is invariant. Moreover we give some conditions so that $F^{\ast }(\mathbb{C}[t_1,\mathrm{.}\mathrm{.}\mathrm{.},t_{n-1}])$ is the ring of invariants of $\varphi$.