Polluted white dwarfs are generally accreting terrestrial-like material that may originate from a debris belt like the asteroid belt in the solar system. The fraction of white dwarfs that are polluted drops off significantly for white dwarfs with masses $M_{rm WD}gtrsim 0.8,rm M_odot$. This implies that asteroid belts and planetary systems around main-sequence stars with mass $M_{rm MS}gtrsim 3,rm M_odot$ may not form because of the intense radiation from the star. This is in agreement with current debris disc and exoplanet observations. The fraction of white dwarfs that show pollution also drops off significantly for low mass white dwarfs $(M_{rm WD}lesssim 0.55,rm M_odot)$. However, the low-mass white dwarfs that do show pollution are not currently accreting but have accreted in the past. We suggest that asteroid belts around main sequence stars with masses $M_{rm MS}lesssim 2,rm M_odot$ are not likely to survive the stellar evolution process. The destruction likely occurs during the AGB phase and could be the result of interactions of the asteroids with the stellar wind, the high radiation or, for the lowest mass stars that have an unusually close-in asteroid belt, scattering during the tidal orbital decay of the inner planetary system.
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