  We compute the gluon polarization tensor in a thermo-magnetic environment in
the strong magnetic field limit at zero and high temperature. The magnetic
field effects are introduced using Schwinger's proper time method. Thermal
effects are computed in the HTL approximation. At zero temperature, we
reproduce the well-known result whereby for a non-vanishing quark mass, the
polarization tensor reduces to the parallel structure and its coefficient
develops an imaginary part corresponding to the threshold for quark-antiquark
pair production. This coefficient is infrared finite and simplifies
considerably when the quark mass vanishes. Keeping always the field strength as
the largest energy scale, in the high temperature regime we analyze two
complementary hierarchies of scales: $q^2\ll m_f^2\ll T^2$ and $m_f^2\ll q^2\ll
T^2$. In the latter, we show that the polarization tensor is infrared finite as
$m_f$ goes to zero. In the former, we discuss the thermal corrections to the
magnetic Debye mass.
