2016/02/12 by Abbott, John, Bigatti, Anna Maria, Robbiano, Lorenzo
#13-04 #13P10 #13P25 #14Q10 #68W30 #Commutative Algebra (math.AC) #FOS: Mathematics
paper · doi:10.48550/arxiv.1602.03993
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 truncated, homogeneous Gröbner basis. The other algorithm, "Direct", computes the implicitization directly using an approach inspired by the generalized Buchberger-Möller 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.