First I tried soaking the pot overnight with dish soap and water. While that loosened up the residue, it did nothing for the burned bottom of the pot. The scraped knuckle I got scouring the bottom didn’t make much of a dent, either.
And then it occurred to me that when I have burnt on patches on my smooth top stove, the first thing I reach for is my stove top cleaner. I happen to use Weiman's Glass Cook Top Cleaner because I can get it for a good (read cheap) price at Walmart, but I'm sure there are other products that are just as good. But it turns out this cream glass top stove cleaner is just as brilliant at cleaning burnt pots!
First, remove all chunks and food bits from the bottom of the pot. Rinse clean and wipe down with a paper towel. Then smear a nice layer of the cream cleaner over the entire burned surface and let it sit for about an hour. At this point, you can use the same scrubbing sponge you use for your stove top and scrub away at the burned pot. If all the black doesn’t come off the first time, simply add more cleaner, let it sit for another hour, and try it again. Be sure to wash your pot well with hot soapy water before using it to remove all traces of the cleaner. And voila! Your pot is as shiny as it was the moment before you burned that crazy soup in the first place!