Carrots are a really nutritious vegetable. They have a fairly high Glycemic Index (GI) #, so they are excluded from the list for Phase 1
. But they are allowed
(and encouraged!) in Phases 2 & 3.
Corn is only allowed in very minute amounts. The way it's listed in the GFGCG makes it sound like you could have one serving once a month or so (once you're in Phase 2). It's very high in sugars, so it's best to avoid it as much as you can.