Algorithms for computing strong μ-bases for rational tensor product surfaces

Implicitizing rational surfaces is a fundamental computational task in Algorithmic Algebraic Geometry. Although the resultant of a μ-basis for a rational surface is guaranteed to contain the implicit equation of the surface as a factor, this resultant may also contain extraneous factors. Moreover, μ-bases for rational surfaces are, in general, notoriously difficult to compute. Here we develop fast algorithms to find μ-bases for rational tensor product surfaces whose resultants are guaranteed to be the implicit equation of the corresponding rational surface with no extraneous factors. We call these μ-bases strong μ-bases. Surfaces with strong μ-bases are relatively rare. We show how these strong μ-bases are related to the number of base points counting multiplicity of the corresponding surface parametrization. In addition, when the base points are simple, we provide tables of rational tensor product surfaces with strong μ-bases based on the bidegree of the rational surface and the number of base points of the parametrization. The bidegrees of the corresponding strong μ-bases are also listed in these tables.