Corrected entry: The film establishes that the core of Mars is made of ice. When planets are coalescing from gas clouds, cumulative gravity draws the heavier materials into the center of the planet. This is why Earth has a nickel-iron core and even gas giants have some solid material in the center. Ice is much too light to occupy this niche in Mars and would have been crushed and replaced by heavier metals in the shell around it, even if the ice was inserted after planetary formation by aliens.
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Correction: Quaid (Arnold) says this, but I wouldn't say that means that "the film establishes it". Quaid is not a scientist and has no idea how this thing works. He sees a big patch of ice in a cave and decides the whole core of Mars is ice, it's probably just one ancient ocean that drained into an underground cave system and froze.
Myridon