How should I support upright posts on a slab with no footer?
First of all, the ground directly under the center of your house is unlikely to freeze, because the house is sitting on top of it protecting it from direct exposure to the cold.  Freezing under the foundation is mainly a concern at the perimeter.

In order to support the ridge beam of your roof, which means the weight of the entire roof, you definitely need to carry that load all the way from the beam down to a footer.

So - directly below where you want the support to connect to the beam, using a plum line mark the foundation.  Cut through the foundation and down to a depth of at least 2 feet (whatever code requires).  Install a new footer in the ground and level the new concrete with the level of the slab.