Global injectivity of planar non-singular maps that are polynomial in one variable

We consider non-singular maps whose components are polynomial in the variable $y$. We prove that if a map has $y$-degree 1, then it is the composition of a triangular map and a quasi-triangular map. We also prove that non-singular $y$-quadratic maps are injective if one of the leading functional coefficients does not vanish. Moreover, $y$-quadratic maps with constant Jacobian determinant are shown to be the composition of a quasi-triangular map and three triangular maps. Other results are given for wider classes of non-singular maps, considering also injectivity on vertical strips $I \times \mathbb R$.