We introduce the notion of radical parametrization of a surface, and we
provide algorithms to compute such type of parametrizations for families of
surfaces, like: Fermat surfaces, surfaces with a high multiplicity (at least
the degree minus 4) singularity, all irreducible surfaces of degree at most 5,
all irreducible singular surfaces of degree 6, and surfaces containing a pencil
of low-genus curves. In addition, we prove that radical parametrizations are
preserved under certain type of geometric constructions that include offset and
conchoids.Comment: 31 pages, 7 color figures. v2: added another case of genus