Poor but Proud is correct. Last Christmas I was making fudge and the pan overflowed. The sugar ate away at the top and now there is a place that cannot be cleaned because it is pitted. Replacing one of these tops is very expensive, I am told. I would never put one of these glass top stoves in a rented place.
I too have a smooth top stove (white, unfortunately), but have found that some elbow grease, Soft Scrub and Scotch Brite work pretty well to get 'most' of the stain off. There may still be a bit of discoloration, but for the most part your stove should look pretty good.
If it's like my friends, it's not burned but pitted. Look very carefully at the deepest angle you can, like almost laying your head on the surface. If you can run your fingers across it and it's a hole and not a bump, stop searching.
You can't fix a pitt. There is nothing you can do but charge them for their neglect and damage. Now, the caviot to that is I am not an expert at these tops so if you can show a photo to a salesperson, they might be able to help.
Good Luck. PBP
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