Torsion of the symmetric algebra and implicitization

Abstract

Recently, a method to compute the implicit equation of a parametrized hypersurface has been developed by the authors. We address here some questions related to this method. First, we prove that the degree estimate for the stabilization of the MacRae’s invariant of Sym A ⁡ ( I ) ν \operatorname {Sym}_{A}(I)_{\nu } is optimal. Then, we show that the extraneous factor that may appear in the process splits into a product of linear forms in the algebraic closure of the base field, each linear form being associated to a non-complete intersection base point. Finally, we make a link between this method and a resultant computation for the case of rational plane curves and space surfaces.