I'm not an expert on climatology, but it doesn't make much sense that there would be more ice at the poles. If it is water ice it should accumulate everywhere because Mars is freezing everywhere. Below are the daytime and night-time surface temperature maps of Mars. You can see that at night in the south polar region it reaches -120 C., and the warmest area is the north pole where the temperature only reaches -65 C. During the day the surface temperature is between -120 C, and +20 C. This may seem encouraging but the last image shows the atmospheric temperature which is clearly in the -80 C range or less, everywhere, all the time, even in areas where the soil temperature is much warmer. There would be a thin boundary layer where the air exchanged heat with the soil via convection, and the temperature would approach that of the soil. But I don't think moisture in the Martian atmosphere would escape the thin boundary layer hugging the ground. It would form a fog perhaps, and remain trapped by hills and craters in low lying areas, only to precipitate out at night when the ground cooled.

I think it would take a different chemical, with a much lower boiling point, to accumulate as frozen precipitate in those hydrogen-rich areas, while remaining gaseous in the mid-latitudes. For example Ammonia (NH3), a colorless gas, has melting and boiling points of , -77.7°C and -33.5°C, respectively. Voila, it would indeed be gaseous in the mid-latitudes and frozen in the south polar region.

Nitrogen oxide (N2O) also has freezing and boiling points of, -90.8 C and -88.5 C, respectively. So it would also accumulate at the south pole of Mars, forming a liquid that might mix & react to form ammonia. If you want more detailed opinions on what might occur, try the usenet sci.chem.