  It is well known that an implicit equation of the offset to a rational planar
curve can be computed by removing the extraneous components of the resultant of
two certain polynomials computed from the parametrization of the curve.
Furthermore, it is also well known that the implicit equation provided by the
non-extraneous component of this resultant has at most two irreducible factors.
In this paper, we complete the algebraic description of this resultant by
showing that the multiplicity of the factors corresponding to the offset can be
computed in advance. In particular, when the parametrization is proper, i.e.
when the curve is just traced once by the parametrization, we prove that any
factor corresponding to a simple component of the offset has multiplicity 1,
while the factor corresponding to the special component, if any, has
multiplicity 2. Hence, if the parametrization is proper and there is no special
component, the non-extraneous part of the resultant is square-free. In fact,
this condition is proven to be also sufficient. Additionally, this result
provides a simple test to check whether or not a given rational curve is the
offset of another curve.
