  Context: Pollux is considered as an archetype of a giant star hosting a
planet. We then discovered a weak magnetic field at its surface using
spectropolarimetry. Aims and Methods: We followed up our investigations on
Pollux first using ESPaDOnS at CFHT and then Narval at TBL to obtain Stokes I
and Stokes V spectra to study their variations for a duration of 4.25 years,
that is, for more than two periods of about 590 d of the RV variations. We used
the least-squares deconvolution (LSD) profiles to measure the longitudinal
magnetic field and to perform a Zeeman Doppler imaging (ZDI) investigation.
Results: The longitudinal magnetic field of Pollux is found to vary with a
sinusoidal behavior and a period similar to that of the RV variations. From the
ZDI investigation a rotation period of Pollux is determined to be equal to
660+/-15 days and possibly different than the period of variations of the RV.
As to the magnetic topology, the poloidal component is dominant and almost
purely dipolar with an inclination of 10.5{\deg} of the dipole with respect to
the rotation axis. The mean strength of the surface magnetic field is 0.44 G.
Conclusions: As to the origin of the magnetic field of Pollux, we favor the
hypothesis that it is maintained through contemporaneous dynamo action. Pollux
appears as the representative of a class of slowly rotating and weakly magnetic
G-K red giants. To explain the sinusoidal RV variations of Pollux, two
scenarios are proposed. If the RV period is different from the rotation period,
the observed periodic RV variations are due to the hosted planet and the
contribution of Pollux magnetic activity is not significantly detected. In the
peculiar case in which the two periods are equal, we cannot discard the
possibility that the activity of Pollux could explain the total RV variations
and that the planet hypothesis would appear unnecessary.
