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

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