We present new, practical algorithms for the hypersurface implicitization problem: namely, given a parametric description (in terms of polynomials or rational functions) of the hypersurface, find its implicit equation. Two of them are for polynomial parametrizations: one algorithm, ElimTH, has as main step the computation of an elimination ideal via a textit{truncated, homogeneous} Grobner basis. The other algorithm, Direct, computes the implicitization directly using an approach inspired by the generalized Buchberger-Moller algorithm. Either may be used inside the third algorithm, RatPar, to deal with parametrizations by rational functions. Finally we show how these algorithms can be used in a modular approach, algorithm ModImplicit, for avoiding the high costs of arithmetic with rational numbers. We exhibit experimental timings to show the practical efficiency of our new algorithms.
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