Compactifications of rational maps, and the implicit equations of their images

In this paper we give different compactifications for the domain and the codomain of an affine rational map $f$ which parametrizes a hypersurface. We show that the closure of the image of this map (with possibly some other extra hypersurfaces) can be represented by a matrix of linear syzygies. We compactify ${\mathbb{A}}^{n-1}$ into an $(n-1)$-dimensional projective arithmetically Cohen-Macaulay subscheme of some ${\mathbb{P}}^N$. One particular interesting compactification of ${\mathbb{A}}^{n-1}$ is the toric variety associated to the Newton polytope of the polynomials defining $f$. We consider two different compactifications for the codomain of $f$: ${\mathbb{P}}^n$ and $({\mathbb{P}}^1{)}^n$. In both cases we give sufficient conditions, in terms of the nature of the base locus of the map, for getting a matrix representation of its closed image, without involving extra hypersurfaces. This constitutes a direct generalization of the corresponding results established in [BuseJouanolou03], [BuseChardinJouanolou06], [BuseDohm07], [BotbolDickensteinDohm09] and [Botbol09].