Flooring Tile - ceramic tile over vinyl

I am installing ceramic tile in a kitchen. It is a concrete slab covered with a vinyl flooring that is 21 years old. It is glued everywhere to the slab, not just the edges. I have a few corners where the vinyl is curled up some. My question is am I better off to glue the curled pieces down, or cut them off, in order to keep the floor warmer, or should I tear up the floor and lay the tile on the concrete. I live in Minnesota. I know there are tile heating systems but do not want to do that.
Thanks for any info.
Vicki

Cut and scrap off any loose vinyl. If the majority or the vinyl is bonded tight , just go over it. You might need to patch in order to keep level. Either way you go, on the concrete or the vinyl, use a multi purpose flexable thinset. The flexable thinset has extra stickum.